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A note on zero commutative and duo rings
... Following Cohn [14], R is said to be reversible if ab = 0 implies ba = 0, for a, b ∈ R. Commutative rings and reduced rings are clearly reversible. Reversible rings were studied under the name zero commutative by Habeb [20]. As a matter of fact, the class of NI rings contains nil rings and reversible rings. ...
... We know that every reduced ring is symmetric [65, Lemma 1.1], but the converse does not hold [2,Example II.5]. Bell [7] used the term Insertionof-Factors-Property (IFP) for a ring R if ab = 0 implies aRb = 0, for a, b ∈ R (Narbonne [53] and Habeb [20] used the terms semicommutative and zero insertive, respectively). Some results about IFP rings are due to Shin [65]. ...
In this paper, we study the reflexive-nilpotents-property (briefly, RNP) for skew PBW extensions. With this aim, we introduce the Σ-skew CN and Σ-skew reflexive (RNP) rings. Under conditions of compatibility, we investigate the transfer of the reflexive-nilpotents-property from a ring of coefficients to a skew PBW extension. We also consider this property for localizations on these families of noncommutative rings. Our results extend those corresponding presented by Bhattacharjee [9].
... On the diameter of the zero-divisor graph over skew PBW extensions Following Lambek [43], a right ideal I of a ring R is called symmetric if abc ∈ I implies acb ∈ I for all a, b, c ∈ R, so we shall call R symmetric if abc = 0 implies acb = 0 for all a, b, c ∈ R; while Anderson-Camillo [9] took the term ZC 3 for this notion. In [21], Cohn defined a ring R as reversible, if ab = 0 implies ba = 0, for every a, b ∈ R. Prior to Cohn's work, reversible rings have been studied under the name completely reflexive by Mason in '[47] and under the name zero commutative, or zc, by Habeb in [27]. Tuganbaev [66] in his monograph on distributive lattices arising in ring theory, investigated a property called commutative at zero, which is equivalent to the reversible condition on rings. ...
... Note that if R is a reversible ring, then Z l (R) = Z r (R) = Z(R). Also, if R is a reversible ring and a ∈ R, then l R (a) = r R (a) is an ideal of R. According to Bell [18], a ring R is called to satisfy the Insertion-of-Factors-Property if ab = 0 implies aRb = 0 for a, b ∈ R. In the literature these rings also go by the names semicommutative, SI and zero insertive due to Narbonne [49], Shin [65] and Habeb [27], respectively. In this paper, we choose "a semicommutative ring" among them, so as to cohere with other related references. ...
The aim of this paper is to investigate the interplay between the algebraic properties of a skew Poincaré–Birkhoff–Witt extesion ring A = σ(R)〈x1,…,xn〉 and the graph-theoretic properties of its zero-divisor graph. We are interested in studying the diameter of the zero-divisor graph of skew PBW extension rings. Among other results, we give a complete characterization of the possible diameters of Γ(A) in terms of the diameter of Γ(R).
... Following Cohn [14], R is said to be reversible if ab = 0 implies ba = 0, for a, b ∈ R. Commutative rings and reduced rings are clearly reversible. Reversible rings were studied under the name zero commutative by Habeb [20]. As a matter of fact, the class of NI rings contains nil rings and reversible rings. ...
... We know that every reduced ring is symmetric [66, Lemma 1.1], but the converse does not hold [2,Example II.5]. Bell [7] used the term Insertion-of-Factors-Property (IFP) for a ring R if ab = 0 implies aRb = 0, for a, b ∈ R (Narbonne [56] and Habeb [20] used the terms semicommutative and zero insertive, respectively). Some results about IFP rings are due to Shin [66]. ...
In this paper, we study the reflexive-nilpotents-property (briefly, RNP) for Ore extensions of injective type, and more generally, skew PBW extensions. With this aim, we introduce the notions of -skew CN rings and -skew reflexive (RNP) rings, for a finite family of ring endomorphisms of a ring R. Under certain conditions of compatibility, we study the transfer of the -skew RNP property from a ring of coefficients to an Ore extension or skew PBW extension over this ring. We also consider this property for localizations of these noncommutative rings. Our results extend those corresponding presented by Bhattacharjee \cite{Bhattacharjee2020}.
... According to Cohn [14], a ring R is called reversible if ab = 0 implies ba = 0, for a, b ∈ R. Prior to Cohn's work, reversible rings were studied under the name completely reflexive by Mason in [36] and under the name zero commutative, or zc, by Habeb in [18]. In his monograph [44] on distributive lattices arising in ring theory, Tuganbaev investigates a property called commutative at zero, which is equivalent to the reversible condition on rings. ...
... Note that if R is a reversible ring, then Z l (R) = Z r (R) = Z(R). Also, if R is a reversible ring and a ∈ R, then l R (a) = r R (a) is an ideal of R. Due to Bell [13], a ring R is called to satisfy the Insertion-of-Factors-Property if ab = 0 implies aRb = 0 for a, b ∈ R. In the literature these rings also go by the names semicommutative, SI and zero insertive due to Narbonne [37], Shin [42] and Habeb [18], respectively. In this note, we choose "a semicommutative ring" among them, so as to cohere with other related references. ...
Let R be an associative ring with nonzero identity. The zero-divisor graph Γ(R) of R is the (undirected) graph with vertices the nonzero zero-divisors of R, and distinct vertices a and b are adjacent if and only if ab = 0 or ba = 0. Let rR(a) and lR(a) be the set of all right annihilators and the set of all left annihilator of an element a ∈ R, respectively, and let annR(a) = lR(a) ∪ rR(a). The relation on R given by a ∼ b if and only if annR(a) = annR(b) is an equivalence relation. The compressed zero-divisor graph ΓE(R) of R is the (undirected) graph with vertices the equivalence classes induced by ∼ other than [0]R and [1]R, and distinct vertices [a]R and [b]R are adjacent if and only if ab = 0 or ba = 0. The goal of our paper is to study the diameter of zero-divisor and the compressed zero-divisor graph of skew Laurent polynomial rings over noncommutative rings. We give a complete characterization of the possible diameters of Γ(R[x,x−1; α]) and ΓE(R[x,x−1; α]), where the base ring R is reversible and also has the α-compatible property.
... According to Cohn [5], a ring R is called reversible if ab = 0 implies ba = 0, for a, b ∈ R. Prior to Cohn's work, reversible rings were studied under the name completely reflexive by Mason in [23] and under the name zero commutative, or zc, by Habeb in [11]. In his monograph [29] on distributive lattices arising in ring theory, Tuganbaev investigates a property called commutative at zero, which is equivalent to the reversible condition on rings. ...
... Note that for the class of reversible rings the set of all left annihilators of any element a ∈ R coincides with the set of its all right annihilators and we denote it by ann R (a). Due to Bell [2], a ring R is called to satisfy the Insertion-of-Factors-Property if ab = 0 implies aRb = 0 for a, b ∈ R. In the literature, these rings also goes by the names semicommutative, SI and zero insertive due to Narbonne [26], Shin [28] and Habeb [11], respectively. In this note, we choose "a semicommutative ring" among them, so as to cohere with other related references. ...
It is well known that a polynomial f(x) over a commutative ring R with identity is a zero-divisor in R[x] if and only if f(x) has a non-zero annihilator in the base ring, where R[x] is the polynomial ring with indeterminate x over R. But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring R[[x]] over an associative non-commutative ring R. We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly AB rings and rings with right Property (A). We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [1 Question, p. 16].
... Recall that a ring is reduced if it has no nonzero nilpotent elements. Lambek [9] called a ring R symmetric provided abc = 0 implies acb = 0 for a, b, c ∈ R. Habeb [10] called a ring R zero commutative if R satisfies the condition: ab = 0 implies ba = 0 for a, b ∈ R, while Cohn [14] used the term reversible for what is called zero commutative. A generalization of a reversible ring is a semicommutative ring. ...
... He proved that (i) R is semicommutative if and only if r R (a) is an ideal of R where r R (a) = {b ∈ R | ab = 0} [6, Lemma 1.2], (ii) every reduced ring is symmetric [6, Lemma 1.1] (but the converse does not hold [3, Example II.5]), and (iii) any symmetric ring is semicommutative but the converse does not hold ([6, Proposition 1.4 and Example 5.4(a)]). Semicommutative rings were also studied under the name zero insertive by Habeb [10]. In [12], Kim and Lee showed that polynomial rings over reversible rings need not be reversible. ...
For a monoid M, we introduce strongly semicommutative rings relative to M, which are a generalization of strongly semicommutative rings, and investigates its properties. We show that every reduced ring is strongly M-semicommutative for any unique product monoid M. Also it is shown that for a monoid M and an ideal I of R. If I is a reduced ring and R/I is strongly M-semicommutative, then R is strongly M-semicommutative.
... In this section, we study graded classical weakly prime submodules over Duo graded rings. A ring A is said to be a left Duo ring if every left ideal of A is a two sided ideal [9,13]. It is obvious that if A is a left Duo ring, then xA ⊆ Ax, for all x ∈ A. Ì ÓÖ Ñ 3.1º Let A be a left Duo graded ring, M be a graded A-module and K be a graded classical weakly prime A-submodule of M. If x, y ∈ h(A) and m ∈ h(M ) such that 0 = xym ∈ K, then either xm ∈ K or ym ∈ K. ...
The goal of this article is to propose and examine the notion of graded classical weakly prime submodules over non-commutative graded rings which is a generalization of the concept of graded classical weakly prime submodules over commutative graded rings. We investigate the structure of these types of submodules in various categories of graded modules.
... cP-Baer skew Laurent polynomial rings. Recall that, a ring R is semicommutative if ab = 0 implies aRb = 0 for a, b ∈ R (this ring is also called a zero insertion(ZI) ring in [14], [23] and [26]). Semicommutative rings have also been called IF P rings in the literature. ...
A ring R is called right -Baer if the right annihilator of a cyclic projective right R-module in R is generated by an idempotent. These rings are a generalization of the right p.q.-Baer rings and abelian rings. Following Birkenmeier and Heider (Commun Algebra 47(3):1348–1375, 2019 https://doi.org/10.1080/00927872.2018.1506462), we investigate the transfer of the -Baer property between a ring R and many polynomial extensions (including skew polynomials, skew Laurent polynomials, skew power series, skew inverse Laurent series), and monoid rings. As a consequence, we answer a question posed by Birkenmeier and Heider (2019).
... We denote Nil(R), the set of nilpotent elements of R. We recall that a ring is said to be reduced whenever it has no non zero nilpotent elements. Again a ring is defined as symmetric in [1] whenever xyz = 0 ⇒ xzy = 0 for any x, y, z ∈ R. In 1999, Cohn [2] defined that a ring is said to be reversible if xy = 0 implies yx = 0 for any x, y ∈ R. Again a ring is called semicommutative if for any x, y ∈ R, xy = 0 implies xRy = 0, this ring is also called ZI ring in [14]. If a ring is commutative, then it is always reversible, symmetric and semicommutative. ...
For a ring endomorphism a, we introduce weakly a-shifting ring which is an extension of reduced as well as a-shifting ring. The notion of weakly a-shifting ring is a generalization of weak a-compatible ring. We investigate various properties of this ring including some kinds of examples in the process of development of this
new concept.
... Then J(R) ⊆ R qnil , N(R) ⊆ R qnil and R qnil ∩ U(R) = 0. It is noted in [7,17,20] that quasinilpotents play an important role in Banach algebras. It is given in [19] that R qnil = {r ∈ R | lim ‖r n ‖ 1 n = 0} = {r ∈ R | λ1 − r ∈ U(R) for 0 ̸ = λ ∈ ℂ}. ...
The reversible property of rings was introduced by Cohn and has important generalizations in noncommutative ring theory. In this paper, reversibility of rings is investigated in relation with quasinilpotents and idempotents, and our argument is spread out based on this. We call a ring R Qnil e -reversible if for any a , b ∈ R {a,b\in R} , being a b = 0 {ab=0} implies b a e ∈ R qnil {bae\in R^{\rm qnil}} for a prescribed idempotent e ∈ R {e\in R} , where R qnil {R^{\rm qnil}} denotes the set of all quasinilpotent elements of R . In the first, we determine the set R qnil {R^{\rm qnil}} for some classes of rings to investigate the structure of Qnil e -reversible rings. In the second, we use R qnil {R^{\rm qnil}} to define Qnil e -reversibility of rings. The notion of Qnil e -reversible ring is a proper generalization of that of e -semicommutative ring, Qnil-semicommutative ring, e -reversible ring and right (left) quasi-duo ring. We obtain some relations between a ring and its quotient rings in terms of Qnil e -reversibility. Applications via some ring extensions and examples illustrating our results are provided.
... In this section, we study graded classical weakly prime submodules over Duo graded rings. A ring A is said to be a left Duo ring if every left ideal of A is a two sided ideal [9,13]. It is obvious that if A is a left Duo ring, then xA ⊆ Ax, for all x ∈ A. ...
The goal of this article is to propose and examine the notion of graded classical weakly prime submodules over non-commutative graded rings which is a generalization of the concept of graded classical weakly prime submodules over commutative graded rings. We investigate the structure of these types of submodules in various categories of graded modules.
... It is clear that for any local ring, strongly (co-)Köthe concept is equivalent to very strongly (co-)Köthe . Then proof is by [21,Proposition 3], Theorem 5.5, Theorem 5.3, Theorem 4.3 and the fact that in any Artinian local ring the Jacobson radical is the unique maximal (prime) ideal. ...
The classical K\"othe's problem posed by G. K\"othe in 1935 asks to describe the rings R such that every left R-module is a direct sum of cyclic modules (are known as left K\"othe rings). K\"othe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved the K\"othe's problem for basic fnite-dimensional algebras. But, so far, the K\"othe's problem was open in the non-commutative setting. Recently, in the paper ["Several Characterizations of Left K\"othe Rings", submitted], we brook the class of left K\"othe rings into three categories of nested: left K\"othe rings, strongly left K\"othe rings and very strongly left K\"othe rings, and then, we solved the K\"othe's problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duality of these notions as co-K\"othe rings, left co-K\"othe rings and strongly left co-K\"othe rings, and then, we give several structural characterizations for each of them.
... A ring R is called a reduced ring if N (R) = (0), i.e., if whenever a 2 = 0, for a ∈ R, then a = 0. A ring R is called a zero commutative ring (zc) if whenever ab = 0, then ba = 0, and R is called a zero insertive ring (zi) if whenever ab = 0, then arb = 0 for all r ∈ R. Reduced rings are zc and every zc ring is zi (see, for example, [4] and [9]). ...
... However the converse of the assertion is false, i.e., there exists a weak symmetric ring which is not symmetric [43,Example 2.2]. Habeb [20] called a ring R zero commutative if for a, b ∈ R, ab = 0 implies ba = 0 (Cohn [15] used the term reversible for what is called zero commutative). A generalization of reversible rings is the notion of semicommutative ring. ...
In this paper, we introduce the concept of -semicommutative ring, for a finite family of endomorphisms of a ring R. We relate this class of rings with other classes of rings such that Abelian, reduced, -rigid, nil-reversible and rings satisfying the -skew reflexive nilpotent property. Also, we study some ring-theoretical properties of skew PBW extensions over -semicommutative rings. We prove that if a ring R is -semicommutative with certain conditions of compatibility on derivations, then for every skew PBW extension A over R, R is Baer if and only if R is quasi-Baer, and equivalently, A is quasi-Baer if and only if A is Baer. Finally, we consider some topological conditions for skew PBW extensions over -semicommutative rings.
... Y(R) (Z(R)) is a right (left) ideal of R, which is called the right (left) singular ideals of R. R is called right (left) non-singular if Y(R) = (0) (Z(R) = (0)) and it is called semi-primitive [8] if J(R) = (0). R is called reduced [8] if it contains no non-zero nilpotent elements, or equivalently, a 2 = 0 implies a = 0, for all a R and it is called a zero insertive (briefly, ZI) ring [11] if for any a , b R, ab = 0 implies that aRb = 0. R is called semi-prime [7] if it contains no non-zero nilpotent ideal. An ideal I of a ring R is said to be a nil ideal [7] if every element of I is a nilpotent element. ...
The concept of generalized flatness and generalized SF-rings were first introduced by the author in 2007. Now, in the present paper we continue to study generalized flatness, generalized SF-rings and determine more properties of them. Moreover, several results are proved. Finally, compare them with regular rings, left non-singular, strongly regular,
strongly π-regular, π-biregular and sπ-weakly regular rings under some conditions.
... Anderson and Camillo [3] studied the rings whose zero products commute, and used the term ZC2 for what is called reversible. Prior to Cohn's work, reversible rings were studied under the names of completely reflexive and zero commutative by Mason [17] and Habe [8], respectively. Tuganbaev [18] investigated reversible rings under the name of commutative at zero. ...
... Narbonne called it semicommutative (see reference in [13]) and many others followed this term (e.g., [18] [19]). Habeb in [20] called it zero insertive. We prefer to call it an AI-rng due to the fact that in this rng all "Annihilators", left or right, are "Ideals". ...
... Due to Birkenmeier et al. [5], a ring is said to be -primal if . Every semicommutative ring isprimal , Semicommutative rings also studied under the name zero insertive by Habeb [6]. Many of authors have been written on ARM property [7] and [8]. ...
This paper investigate the possibility of inheriting the properties of the ring
... A ring R is called reduced if it has no nonzero nilpotent element. According to Cohn [6], a ring R is called reversible if ab = 0 implies ba = 0 for a, b ∈ R. Prior to Cohn's work, reversible rings were studied under the name completely reflexive by Mason in [17] and under the name zero commutative, or zc, by Habeb in [10]. In his monograph [27] on distributive lattices arising in ring theory, Tuganbaev investigates a property called commutative at zero, which is equivalent to the reversible condition on rings. ...
Given a ring R, a strictly totally ordered monoid (S,) and a monoid homomorphism ω:S→End(R), one can construct the skew generalized power series ring R[[S,ω,{precedes above single-line equals sign}]], consisting all of the functions from a monoid S to a coefficient ring R whose support is artinian and narrow, where the addition is pointwise, and the multiplication is given by convolution twisted by an action ω of the monoid S on the ring R. In this paper, we consider the problem of determining some annihilator and zero-divisor properties of the skew generalized power series ring R[[S,ω,]] over an associative non-commutative ring R. Providing many examples, we investigate relations between McCoy property of skew generalized power series ring, namely (S,ω)-McCoy property, and other standard ring-theoretic properties. We show that if R is a local ring such that its Jacobson radical J(R) is nilpotent, then R is (S,ω)-McCoy. Also if R is a semicommutative semiregular ring such that J(R) is nilpotent, then R is (S,ω)-McCoy ring. © 2019, Mathematical Society of the Rep. of China. All rights reserved.
... Throughout this paper, a ring means an associative ring with identity. We write U (R) for the set of all units in R, T n (R) stands for the ring of all n × n triangular matrices over a ring R. A ring R is called semicommutative if for any a, b ∈ R, ab = 0 implies aRb = 0, this ring is also called ZI ring in [9] and [13], while, in [24], R is said to be central semicommutative if ab = 0 implies aRb is central in R. And in [17] a ring R is called weakly semicommutative, if for any a, b ∈ R, ab = 0 implies arb is a nilpotent element for each r ∈ R. Another generalization is made in [6], in which a ring R is called nil-semicommutative-II if a, b ∈ R satisfy ab ∈ Nil(R), then arb ∈ Nil(R) for any r ∈ R where Nil(R) is the set of all nilpotent elements of R. A similar concept is nil-semicommutativity is investigated in [20], in which it is said that a ring R is nil-semicommutative-I if for all nilpotent elements a, b of R, ab = 0 implies aRb = 0. ...
... Anderson and Camillo [3], observed the rings whose zero products commute, and used the term ZC 2 for what is called reversible. Prior to Cohn's work, reversible rings were studied under the names of completely reflexive and zero commutative by Mason [22] and Habeb [9], respectively. While, Tuganbaev [29] investigated reversible rings in his monograph on distributive lattices arising in ring theory, using the name of commutative at zero in place of reversible. ...
The reversible property of rings was initially introduced by Habeb and plays a role in noncommutative ring theory. In this note we study the reversible ring property on nilpotent elements, introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring) as a generalization of reversible rings. We first find the CNZ property of 2 by 2 full matrix rings over fields, which provides a basis for studying the structure of CNZ rings. We next observe various kinds of CNZ rings including ordinary ring extensions.
... Y(R) (Z(R)) is a right (left) ideal of R, which is called the right (left) singular ideals of R. R is called right (left) non-singular if Y(R) = (0) (Z(R) = (0)) and it is called semi-primitive [8] if J(R) = (0). R is called reduced [8] if it contains no non-zero nilpotent elements, or equivalently, a 2 = 0 implies a = 0, for all a R and it is called a zero insertive (briefly, ZI) ring [11] if for any a , b R, ab = 0 implies that aRb = 0. R is called semi-prime [7] if it contains no non-zero nilpotent ideal. ...
The concept of generalized flatness and generalized SF-rings were first introduced by theauthor in 2007. Now, in the present paper we continue to study generalized flatness,generalized SF-rings and determine more properties of them. Moreover, several resultsare proved. Finally, compare them with regular rings, left non-singular, strongly regular,strongly π-regular, π-biregular and sπ-weakly regular rings under some conditions.
... rings with no nonzero nilpotent elements) and commutative rings are reversible. Prior to Cohn's work, reversible rings were studied under the name completely reflexive by Mason in [30] and under the name zero commutative, or zc, by Habeb in [15]. In his monograph [37] on distributive lattices arising in ring theory, Tuganbaev investigates a property called commutative at zero, which is equivalent to the reversible condition on rings. ...
Let R be a ring and (S,≤) a strictly ordered monoid. The construction of generalized power series ring R[[S]] generalizes some ring constructions such as polynomial rings, group rings, power series rings and Mal’cev–Neumann construction. In this paper, for a reversible right Noetherian ring R and a m.a.n.u.p. monoid (S,≤), it is shown that (i) R is power-serieswise S-McCoy, (ii) R[[S]] have Property (A), (iii) R is right zip if and only if R[[S]] is right zip, (iv) R is strongly AB if and only if R[[S]] is strongly AB. Also we study the interplay between ring-theoretical properties of a generalized power series ring R[[S]] and the graph-theoretical properties of its undirected zero divisor graph of Γ(R[[S]]). A complete characterization for the possible diameters Γ(R[[S]]) is given exclusively in terms of the ideals of R. Also, we present some examples to show that the assumption “R is right Noetherian” in our main results is not superfluous.
... Habeb [7] and Mason [18] studied this condition under the names of zero commutative and completely reflexive, respectively; while such rungs were also investigated by Tuganbaev [22], using the name of commutative at zero. On the other hand, Cohn [5] called a ring reversible if the above condition holds. ...
The reversible property an important role in noncommutative ring theory. Recently, the study of the reversible ring property on nilpotent elements is established by A.M. Abdul-Jabbar et al., introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring) as a generalization of reversible rings. We here study this property skewed by a ring endomorphism α, and such ring is called a right α-skew CNZ ring which is an extension of CNZ rings as well as a generalization of right α-skew reversible rings, and then investigate the structure of right α-skew CNZ rings and their related properties. Consequently, several known results are obtained as corollaries of our results.
... According to Cohn [16], a ring R is called reversible if ab = 0 implies that ba = 0 for a, b ∈ R. Prior to Cohns work, reversible rings were studied under the name completely reflexive by Mason in [35] and under the name zero commutative, or zc, by Habeb in [22]. In his monograph [47] on distributive lattices arising in ring theory, Tuganbaev investigates a property called commutative at zero, which is equivalent to the reversible condition on rings. ...
Let (Formula presented.) be an associative ring with identity, (Formula presented.) a monoid and (Formula presented.) a monoid homomorphism. When (Formula presented.) is a u.p.-monoid and (Formula presented.) is a reversible (Formula presented.)-compatible ring, then we observe that (Formula presented.) satisfies a McCoy-type property, in the context of skew monoid ring (Formula presented.). We introduce and study the (Formula presented.)-McCoy condition on (Formula presented.), a generalization of the standard McCoy condition from polynomial rings to skew monoid rings. Several examples of reversible (Formula presented.)-compatible rings and also various examples of (Formula presented.)-McCoy rings are provided. As an application of (Formula presented.)-McCoy rings, we investigate the interplay between the ring-theoretical properties of a general skew monoid ring (Formula presented.) and the graph-theoretical properties of its zero-divisor graph (Formula presented.).
... According to Cohn [7], a ring R is called reversible if ab = 0 implies ba = 0, for a, b ∈ R. Also by [7], a ring R is usually called reduced if it has no nonzero nilpotent elements. Prior to Cohn's work, reversible rings were studied under the name completely reflexive by Mason in [30] and under the name zero commutative, or zc, by Habeb in [12]. In his monograph [37] on distributive lattices arising in ring theory, Tuganbaev investigates a property called commutative at zero, which is equivalent to the reversible condition on rings. ...
The study of rings with right Property (A), has done an important role in noncommutative ring theory. Following literature, a ring R has right Property (A) if every finitely generated two-sided ideal consisting entirely of left zero-divisors has a nonzero right annihilator. Our results in this paper concerns the right Property (A) of Ore extensions as well as skew power series rings. We will show that if R is a right duo ring, then the skew power series ring R[[x; α]] has right Property (A), when R is right Noetherian and α-compatible. Moreover, for a right duo ring R which is (α,δ)-compatible, it is shown that (i) the Ore extension ring R[x; α,δ] has right Property (A) and (ii) R[x; α,δ] is right zip if and only if R is right zip. As a corollary of our results, we provide answers to some open questions related to Property (A), raised in [C. Y. Hong, N. K. Kim, Y. Lee and S. J. Ryu, Rings with Property (A) and their extensions, J. Algebra 315 (2007) 612–628].
... Note that, R is NI if and only if Ni (R) forms a two-sided ideal if and only if R/Ni * (R) is reduced. Due to Birkenmeier et al. [7], a ring R is called 2-primal, if its prime radical contains every nilpotent element of R. Also, due to Bell [6], a ring R is called to satisfy the Insertion-of-Factors-Property (simply, IFP), if ab = 0 implies aRb = 0 for a, b ∈ R. In the literature, this rings also goes by the names semicommutative, SI and zero-insertive due to Narbonne [24], Shin [28] and Habeb [10], respectively. In this note, we choose "a semicommutative ring" among them, so as to cohere with other related references. ...
Armendariz rings are generalization of reduced rings, and therefore, the set of nilpotent elements plays an important role in this class of rings. There are many examples of rings with nonzero nilpotent elements which are Armendariz. Observing structure of the set of all nilpotent elements in the class of Armendariz rings, Antoine introduced the notion of nil-Armendariz rings as a generalization, which are connected to the famous question of Amitsur of whether or not a polynomial ring over a nil coefficient ring is nil. Given an associative ring R and a monoid M, we introduce and study a class of Armendariz-like rings defined by using the properties of upper and lower nilradicals of the monoid ring R[M]. The logical relationship between these and other significant classes of Armendariz-like rings are explicated with several examples. These new classes of rings provide the appropriate setting for obtaining results on radicals of the monoid rings of unique product monoids and also can be used to construct new classes of nil-Armendariz rings. We also classify, which of the standard nilpotence properties on polynomial rings pass to monoid rings. As a consequence, we extend and unify several known results.
... l R (X)) will be denoted by the right(resp. left)annihilator of X in R. In 1990, Habeb studied zero commutative ring in [5]. A ring R is called zero commutative, if ab = 0 implies ba = 0 for any a, b ∈ R. In 1999 [4] used the terminology "reversible ring" instead of "zero commutative ". ...
In this paper, we apply some properties of reversible rings, Baerness of fixed rings, skew group rings and Morita Context rings to get conditions that shows fixed rings, skew group rings and Morita Context rings are reversible. Moreover, we investigate conditions in which Baer rings are reversible and reversible rings are Baer.
... (3) A ring R is called an MERT if every essential maximal right ideal of R is an ideal. (4) R is a left (right) Kasch ring if every maximal left (right) ideal is a left (right) annihilator of R. Motivated by the well known result of Kaplansky (i.e., A commutative ring R is von Neumann regular if and only if every simple R-module is injective), many authors studied rings whose simple (singular) modules are injective (P -injective, GP -injective) (see [1], [2], [4][5][6], [9], [11], [12], [14], [15]). It was proven that: (1) R is strongly regular if and only if R is a left (or right) quasi-duo ring whose simple left R-modules are injective (or P -injective) (see [11]); (2) A ring R is strongly regular if and only if R is a left duo ring whose simple singular left R-modules are P -injective (see [14]); (3) A ring R is strongly regular if and only if R is a left duo ring whose simple singular left R-modules are Y J-injective if and only if R is a left quasi-duo ring whose simple left R-modules are Y J-injective (see [2]); (4) A ring R is strongly regular if and only if R is a weakly right duo ring whose simple singular right Rmodules are right GP -injective (see [6]). ...
Let R be a ring. A right R-module M is PS-injective if every R- homomorphism f: aR → M for every principally small right ideal aR can be extended to R → M. We investigate, in this paper, rings whose simple singular modules are PS- injective. New characterizations of semiprimitive rings and semisimple Artinian rings are given.
... A right (or left) ideal I of a ring R is said to have the IFP if ab ∈ I implies aRb ⊆ I for a, b ∈ R, and we will call a ring IFP if the zero ideal has the IFP. Narbonne [12] and Shin [13] used the terms semicommutative and SI for the IFP, respectively; while, IFP rings were also studied under the name zero insertive by Habeb [7]. IFP rings are 2-primal [13, Theorem 1.5]. ...
We in this note introduce the concept of semi-IFP rings which is a generalization of IFP rings. We study the basic structure of semi-IFP rings, and construct suitable examples to the situations raised naturally in the process. We also show that the semi-IFP does not go up to polynomial rings.
... The concept of symmetric rings was introduced by Lambek [17] to unify sheaf representations of commutative rings and reduced rings. Prior to Cohn's work, reversible rings were studied under the names completely reflexive and zero commutative by Mason [19] and Habeb [11], respectively. Tuganbaev [22] investigated reversible rings in his monograph on distributive lattices arising in ring theory, using the name commutative at zero in place of reversible. ...
We study the connections between idempotents and zero-divisors in several kinds of ring theoretic properties. We next study several ring theoretic properties and examples related to reversible rings.
... Throughout this paper, all rings are associative not necessarily with identity, l, m, n are positive integers and n ≥ 2. Let M l×m (R) and T m (R) denote respectively the set of all l × m matrices and the m × m upper triangular matrix ring over a ring R. A ring R is called reduced if it has no non-zero nilpotent elements. Habeb [3] called a ring R zero commutative if ab = 0 implies ba = 0 for a, b ∈ R, while Cohn [2] used the term reversible for this notion. Anderson and Camillo [1] considered the class of rings R satisfying zero products commuting property for n ≥ 2, i.e., whenever a 1 a 2 · · · a n = 0 for a 1 , a 2 , . . . ...
It is well known that the m×m upper triangular matrix ring over any ring is not ZI n (and so not ZC n ) for m≥2. In this paper, we find some ZC n subrings and ZI n subrings of the upper triangular matrix ring over a reduced ring.
... According to Habeb [9], a ring R is zero commutative if for all a, b ∈ R, ab = 0 implies ba = 0. Cohn [8] used the term reversible for what is called zero commutative. On the other hand, Lambek [18] called a ring R symmetric provided abc = 0 implies acb = 0 for a, b, c ∈ R. We note that reduced rings are symmetric [21, Lemma 1.1], and symmetric rings are clearly zero commutative, but the converse of each of these implications does not hold in general. ...
A ring A is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. Such rings have been extensively studied in literature. For a ring endomorphism α, we introduce the notion of α-Armendariz rings by considering the polynomials in the skew polynomial ring R[x; a] in place of the ring R[x], A number of properties of this generalization are established, and connections of properties of an α-Armendariz ring R with those of the ring R[x; α] are investigated. In particular, among other results, we show that there is a strong connection of the Baer property and the p.p.-property (principal ideals are projective) of the two rings, respectively. Several known results follow as consequences of our results.
... Throughout this paper, all rings are associative with identity. A ring R is called semicommutative if for any a, b ∈ R, ab = 0 implies aRb = 0 (this ring is also called ZI ring in [2,8]). R is semicommutative if and only if any right (left) annihilator over R is an ideal of R by [4,Lemma 1] or [7,Lemma 1.2]. ...
We introduce weakly semicommutative rings which are a general-ization of semicommutative rings, and give some examples which show that weakly semicommutative rings need not be semicommutative. Also we give some relations between semicommutative rings and weakly semicommutative rings.
... Following Bell [1], a ring R is called to satisfy the insertion-of-factorsproperty (simply, an IFP ring) if ab = 0 implies aRb = 0 for a, b ∈ R. Narbonne [10], Shin [11], and Habeb [4] used the terms semicommutative, SI, and zero-insertive for the IFP ring property, respectively. A ring is usually called reduced if it has no nonzero nilpotent elements. ...
The structures of finite local rings of order 16 with nonzero Jacobson radical are investigated. The whole shape of non-commutative local rings of minimal order is completely determined up to isomorphism.
We study classical Köthe’s problem, concerning the structure of non-commutative rings with the property that: “every left module is a direct sum of cyclic modules". In 1934, Köthe showed that left modules over Artinian principal ideal rings are direct sums of cyclic modules. A ring R is called a left Köthe ring if every left R-module is a direct sum of cyclic R-modules. In 1951, Cohen and Kaplansky proved that all commutative Köthe rings are Artinian principal ideal rings. During the years 1961–1965, Kawada solved Köthe’s problem for basic finite-dimensional algebras: Kawada’s theorem characterizes completely those finite-dimensional algebras for which any indecomposable module has a square-free socle and a square-free top, and describes the possible indecomposable modules. But, so far, Köthe’s problem is open in the non-commutative setting. In this paper, we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solve Köthe’s problem by giving several characterizations of these rings in terms of describing the indecomposable modules. Finally, we give a new generalization of Köthe–Cohen–Kaplansky theorem.
In this paper, we introduce the concept of -semicommutative ring for a finite family of endomorphisms of a ring R. We relate this class of rings with other classes of rings such as Abelian, reduced, -rigid, nil-reversible and rings satisfying the -skew reflexive nilpotent property. Also, we study some ring-theoretical properties of skew PBW extensions over -semicommutative rings. We prove that if a ring R is -semicommutative with certain conditions of compatibility on derivations, then for every skew PBW extension A over R, R is Baer if and only if R is quasi-Baer, and equivalently, A is quasi-Baer if and only if A is Baer. Finally, we consider some topological conditions for skew PBW extensions over -semicommutative rings.
In this article, we provide a complete characterization of abelian group rings which are Köthe rings. We also provide characterizations of (possibly non-abelian) group rings over division rings which are Köthe rings, both in characteristic zero and in prime characteristic, and prove a Maschke type result for pure semisimplicity of group rings. Furthermore, we illustrate our results by several examples.
Communicated by Eric Jespers
In this article we provide a complete characterization of abelian group rings which are K\"{o}the rings. We also provide characterizations of (possibly non-abelian) group rings over division rings which are K\"{o}the rings, both in characteristic zero and in prime characteristic, and prove a Maschke type result for pure semisimplicity of group rings. Furthermore, we illustrate our results by several examples.
We solve the classical Köthe's problem, concerning the structure of non-commutative rings with the property that: "every left module is a direct sum of cyclic modules". A ring R is left (resp., right) Köthe if every left (resp., right) R-module is a direct sum of cyclic R-modules. Köthe [Math. Z. 39 (1934), 31-44] showed that all Artinian principal ideal rings are left Köthe rings. Cohen and Kaplansky [Math.Z 54 (1951), 97-101] proved that all commutative Köthe rings are Artinian principal ideal rings. Faith [Math. Ann. 164 (1966), 207-212] characterized all commutative rings whose proper factor rings are Köthe rings. However, Nakayama [Proc. Imp. Acad. Japan 16 (1940), 285-289] gave an example of a left Köthe ring which is not a principal ideal ring. Kawada [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A7 (1962), 154-230; Sect. A8 (1965),165-250] completely solved Köthe's problem for basic finite dimensional algebras. So far the Köthe's problem was still open in the non-commutative setting. In this paper, among other related results, we solve Köthe's problem for any ring. We also determine the structure of left co-Köthe rings (rings whose left modules are direct sums of co-cyclics modules). Finally, as an application, we present several characterizations of left Kawada rings, which generalizes a well-known result of Ringel on finite dimensional Kawada algebras (a ring R is called left Kawada if any ring Morita equivalent to R is a left Köthe ring). * The research of the second author was in part supported by two grants from IPM (No.1400130213), and (No.1401130213). This research is partially carried out in the IPM-Isfahan Branch. †
We solve the classical Kothes problem, concerning the structure of non-commutative rings with the property that: every left module is a direct sum of cyclic modules. A ring R is left (resp., right) Kothe if every left (resp., right) R-module is a direct sum of cyclic R-modules. Kothe [Math. Z. 39 (1934), 31-44] showed that all Artinian principal ideal rings are left Kothe rings. Cohen and Kaplansky [Math.Z 54 (1951), 97-101] proved that all commutative Kothe rings are Artinian principal ideal rings. Faith [Math. Ann. 164 (1966), 207-212] characterized all commutative rings whose proper factor rings are Kothe rings. However, Nakayama [Proc. Imp. Acad. Japan 16 (1940), 285-289] gave an example of a left Kothe ring which is not a principal ideal ring. Kawada [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 7 (1962), 154-230; Sect. A 8 (1965),165-250] completely solved Kothes problem for basic finite dimensional algebras. So far the Kothe's problem was still open in the non-commutative setting. In this paper, among other related results, we solve Kothes problem for any ring. We also determine the structure of left co-Kothe rings (rings whose left modules are direct sums of co-cyclics modules). Finally, as an application, we present several characterizations of left Kawada rings, which generalizes a well-known result of Ringel on finite dimensional Kawada algebras (a ring R is called left Kawada if any ring Morita equivalent to R is a left Kothe ring).
In this paper, we establish a generalization of the class of semicommutative rings which is called * -semicommutative ring. We also discuss the relations among * -semicommutative rings, * -reversible rings and reversible rings. Moreover, we investigate some properties of these rings.
T.K. Kwak and Y. Lee called a ring R satisfy the commutativity of nilpotent elements at zero[1] if ab = 0 for a, b ∈ N(R) implies ba = 0. For simplicity, a ring R is called CNZ if it satisfies the commutativity of nilpotent elements at zero. In this paper we study an extension of a CNZ ring with its endomorphism. An endomorphism α of a ring R is called strong right ( resp., left) CNZ if whenever aα(b) = 0(resp., α(a)b = 0 ) for a, b ∈ N(R) ba = 0. A ring R is called strong right (resp., left) α-CNZ if there exists a strong right (resp., left) CNZ endomorphism α of R, and the ring R is called strong α- CNZ if R is both strong left and right α- CNZ. Characterization of strong α- CNZ rings and their related properties including extensions are investigated . In particular, it’s shown that a ring R is reduced if and only if U2(R) is a CNZ ring. Furthermore extensions of strong α- CNZ rings are studied.
In this paper, we study the notion of McCoy ring over the class of non-commutative rings of polynomial type known as skew Poincaré–Birkhoff–Witt extensions. As a consequence, we generalize several results about this notion considered in the literature for commutative rings and Ore extensions.
This note is intended as a discussion of some generalizations of the notion of a reversible ring, which may be obtained by the restriction of the zero commuta-tive property from the whole ring to some of its subsets. By the INCZ property we will mean the commutativity of idempotent elements of a ring with its nilpo-tent elements at zero, and by ICZ property we will mean the commutativity of idempotent elements of a ring at zero. We will prove that the INCZ property is equivalent to the abelianity even for nonunital rings. Thus the INCZ property implies the ICZ property. Under the assumption on the existence of unit, also the ICZ property implies the INCZ property. As we will see, in the case of nonunital rings, there are a few classes of rings separating the class of INCZ rings from the class of ICZ rings. We will prove that the classes of rings, that will be discussed in this note, are closed under extending to the rings of polynomials and formal power series.
The purpose of this paper is to study the rings in which each simple right R-module is P-injective or flat. Such rings will be called right SPF-rings. We study conditions under which SPF- rings are strongly regular and biregular. Among other results we prove that: 1- If R is a semi-prime, MERT, SPF, P. I ring, then R is a regular ring. 2- If R be a ZC, SPF-ring, then R is biregular.
An endomorphism α of a ring R is called right reversible if whenever ab = 0 for a, b ∈ R, then bα(a) = 0. A ring R is called right α-reversible if there exists a right reversible endomorphism α of R. The notion of an α-reversible ring is a generalization of α-rigid rings as well as an extension of reversible rings. We study characterizations of α-reversible rings and their related properties including extensions. The relationship between α-reversible rings and generalized Armendariz rings is also investigated. Several known results relating to α-rigid and reduced rings can be obtained as corollaries of our results. © 2009 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.
This paper gives the definition of the right(left) quasi-α-reversible rings, and discusses the relationships between the class rings and linearly McCoy, reversible, α-rigid, reduced, Abelian and weak α-skew Armendariz rings. We investigate the related properties and some basic extensions of right (left) quasi-α-reversible rings. Several known results relating the to α-reversible rings and α-rigid rings can be obtained as corollaries of our results.
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