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Pure ideals in commutative reduced Gelfand rings with unity
... The compact elements of Rad(A) are the ideals J = [a 1 ] ∨ · · · ∨ [a n ] for some finitely many a 1 , . . . , a n in A. The top element of Rad(A) is the compact element [1] = A, and the bottom element is [0], which is the zero ideal if A is reduced. ...
... On the other hand, A is called a Gelfand ring if whenever a + b = 1 in A, then (1 + ar)(1 + sb) = 0 for some r, s ∈ A. In [5, Theorem 4.1], Contessa proves (unavoidably using some form of the Axiom of Choice) that pmrings are precisely Gelfand rings. Thus, Al-Ezeh [1] uses the terms interchangeably. Banaschewski [4] though distinguishes between the two, and proves in [4, Proposition 3] that every pm-ring is Gelfand, if one assumes the Prime Ideal Theorem. ...
... In preparation for the characterization, we have in mind, let us note that if I ∨ J = [1] for some I, J ∈ Rad(A), then I + J = A. This simple (but useful) observation already appears in [4]. ...
For a prime ideal P of a commutative ring A with identity, we denote (as usual) by OP its zero-component; that is, the set of members of P that are annihilated by nonmembers of P. We study rings in which OP is an essential ideal, whenever P is an essential prime ideal. We characterize them in terms of the lattices (which are, in fact, complete Heyting algebras) of their radical ideals. We prove that the classical ring of quotients of any ring of this kind is itself of this kind. We show that direct products of rings of this kind are themselves of this kind. We observe that the ring of real-valued continuous functions on a Tychonoff space is of this kind precisely when the underlying set of the space is infinite. Replacing OP with the pure part of P, we obtain a formally stronger variant which is still characterizable in terms of the lattices of radical ideals.
... A large class of commutative rings can be classified through the pure ideals of the ring. The pure ideals in C(X) were completely characterized in [3]. ...
... In fact the answer is not true in general. The following example was given in [3]. Let R be the space of reals. ...
... We will show that intersection of arbitrary family of pure ideals in C(X) is pure if and only if X is basically disconnected, but first we'll need some preliminaries. [3]). An Ideal I of C(X) is pure if and only if I = O A for some closed subset A⊂ βX. ...
Let C(X) be the ring of all continuous real valued functions dened on a completely regular T1 space. For each ideal I in C(X) let mI be the pure part of the ideal I: In this article we show that mI = O (I); where (I) = f2I cl XZ(f). The pure part of many ideals in C(X) is calculated. We found that mCK(X), the pure part of the ideal of functions with compact support, is nitely generated if and only if X- (CK(X)) is com- pact, mCK(X) is countably generated if and only if X- (CK(X)) is Lin- del} o and mCK(X) is generated by a star nite set if and only if X- (CK(X)) is paracompact. Similar results are obtained for the pure part of the ideal C (X), the ideal of functions with pseudocompact support.
... Pure ideals have been studied in various articles in the literature over the years, see e.g. [2], [3], [4], [5], [6], [7] and [8]. In [1], it is shown that a ring is reduced if and only if its Pure and N-pure ideals are the same. ...
... The following result improves [4,Theorem 1.5]. Proof. ...
In this paper, we consider the N-pure notion. An ideal I of a ring R is said to be N-pure, if for every there exists such that , where N(R) is nil radical of R. We provide new characterizations for N-pure ideals. In addition, N-pure ideals of an arbitrary ring are identified. Also, some other properties of N-pure ideals are studied. finally, we prove some results about the endomorphism ring of pure and N-pure ideals.
... An ideal I of a ring R is called pure if I ∩ J = IJ for every ideal J of R, or equivalently, if for any a ∈ I, there exists b ∈ I such that a = ab, see [9,Chap. 7,§1]. Obviously, any sum and any finite intersection of pure ideals is a pure ideal. ...
... (1) ⇒ (7) ⇒ (8) ⇒ (9) Clear. On the other hand, it is known that the pure ideals of C(X) are precisely of the form O A for some closed subset A of βX, see [1]. Applying these results, we infer that every nonzero ideal contains a nonzero pure ideal. ...
In this paper, we describe how intersections with a totality of some ideals affect the essentiality of an ideal. We mainly study intersections with every (a) annihilator ideal, (b) prime ideal (c) strongly irreducible ideal (d) irreducible ideal and every pure ideal. After some general results, the paper focuses on C(X) to characterize spaces X when every irreducible ideal of C(X) is pseudoprime. We also characterize the rings of continuous functions C(X) in which every pseudoprime ideal is strongly irreducible. We give a negative answer to a question raised by Gilmer and McAdam.
... For any element a in R, we define the right annihilater of a by r(a)={x∈R:ax=0} , and likewise the left annihilater l(a) . Y,Z,J will denote respectively the right singular ideal, the left singular ideal and the Jacobson radical of R. Recall that 1) An ideal I is said to be a right (left) pure if for every a∈I , there exists b∈I such that a=ab(ba) [1] 2) R is called a uniform ring if for every non-zero ideal of R is essential , [4] .3) A ring R is said to be left kasch ring, if every maximal right ideal is a right annihilator [3] 4) R is said to be strongly regular if for each a∈R , there exists x∈R such that a=a 2 x. ...
... A ring R is said to be left kasch ring, if every maximal right ideal is a right annihilator [3] 4) R is said to be strongly regular if for each a∈R , there exists x∈R such that a=a 2 x. Following [1] . 5)A ring R is called reduced if R has no non-zero nilpotent element and an ideal I of a ring R is said to be right (left) GP-ideals if for every a∈I, there exists b∈I and a positive integer n such that a n =a n b(ba n ) [5]. ...
MRGP ﺍﻟﻘﺴﻤﺔ ﻭﺤﻠﻘﺔ ﻜﺎﺵ ،ﺤﻠﻘﺔ ﺒﻘﻭﺓ ﺍﻟﻤﻨﺘﻅﻤﺔ ﺍﻟﺤﻠﻘﺔ ﻤﻊ ﻭﻋﻼﻗﺘﻬﺎ ﺨﻭﺍﺼﻬﺎ ﺒﻌﺽ ﻋﺭﺽ ﻭﺘﻡ. ABSTRACT This paper introduces the notion of maximal GP-ideal .We studied the class of rings whose maximal left ideal are right GP-ideal. We call such ring MRGP-rings. We consider a necessary and sufficient condition for MRGP-rings to be MRCP-rings. We also study the connection between MRGP-ring, kasch ring, division ring and the strongly regular ring.
... Omidi S. et al. there exists b ∈ I such that a = ab. The notion of pure ideals in commutative reduced Gelfand rings with unity has been studied in [2]. ...
... A canonical hypergroup is a non-empty set H endowed with an additive hyperoperation + : H × H → P * (H), satisfying the following properties : (1) for any x, y, z ∈ H, x + (y + z) = (x + y) + z; (2) for any x, y ∈ H, x + y = y + x; ...
In this paper, we present some basic notions of simple ordered semihypergroups and regular ordered Krasner hyperrings and prove some results in this respect. In addition, we describe pure hyperideals of ordered Krasner hyperrings and investigate some properties of them. Finally, some results concerning purely prime hyperideals are proved. © 2017 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg
... The references [5][6][7][8] are the first who worked on the concept of pure ideals. This concept has been developed and studied extensively by [9][10][11][12]. Many papers have tackled this notion by different ways. ...
Ring theory is one of the branches of an abstract algebra. This field is the study of a mathematical system with two binary operations. In this branch, many articles have studied this algebraic structure and presented some new works. However, the concept of purity has been studied before more than 40 years ago, especially the relation between the pure ideal and some other ideals on the given ring. In this paper, we survey the important results that concern with pure ideals. Some different types of ideals have been discussed such as N-pure ideals, z-ideals, Π-pure ideals and strongly pure ideals. Moreover, some recent results based on the work of several researchers have been summarized. On the other hand, regarding these types of ideals, some questions have been presented. Furthermore, many important results about various types of rings which are based on the notion of pure ideals have been studied.
... Tus, understanding the algebraic properties and structural characteristics of the Hurwitz series ring is essential. Te primary objective of this paper is to characterize diferential ideals within the Hurwitz series ring HR over the underlying ring R. In ring theory, an ideal I of a ring R is called pure if, for every a ∈ I, there exists a b ∈ I such that ab � a. Pure ideals are essential in ring theory as they help in classifying various types of rings, such as Von Neumann regular rings (VNR-rings) and PF-rings [4,5]. Terefore, characterizing pure ideals in the Hurwitz series rings HR is of signifcant interest. ...
This paper offers an in-depth investigation into pure ideals within the Hurwitz series ring. Specifically, by focusing on the Hurwitz series ring, denoted as HR over a ring R, we present a comprehensive characterization of differential ideals. In this paper, we prove that these differential ideals can be expressed in the form HI, where I represents an ideal in the underlying ring R. Through this analysis, a comprehensive understanding of the structure and properties of pure ideals within the Hurwitz series ring is achieved.
... The class of rings that satisfy this universal property had been investigated by De Marco and Orsatti (1971) under pm-rings. Gelfand rings have been the main subject of many articles in the literature over the years and are still of current interest, see e.g., Johnstone (1982), Contessa (1982), Contessa (1984), Al-Ezeh (1989, 1990a, Banaschewski (2000), Aghajani and Tarizadeh (2020). Motivated by De Marco and Orsatti (1971), the notion of a pm-lattice was introduced by Pawar and Thakare (1977) as a bounded distributive lattice in which any prime ideal is contained in a unique maximal ideal. ...
In this paper, a combination of algebraic and topological methods is applied to obtain new and structural results on Gelfand residuated lattices. It is demonstrated that Gelfand’s residuated lattices strongly tied up with the hull–kernel topology. Particularly, it is shown that a residuated lattice is Gelfand if and only if its prime spectrum, equipped with the hull–kernel topology, is normal. The class of soft residuated lattices is introduced, and it is shown that a residuated lattice is soft if and only if it is Gelfand and semisimple. Gelfand residuated lattices are characterized using the pure part of filters. The relation between pure filters and radicals in a Gelfand residuated lattice is described. It is shown that a residuated lattice is Gelfand if and only if its pure spectrum is homeomorphic to its usual maximal spectrum. The pure filters of a Gelfand residuated lattice are characterized. Finally, it is proved that a residuated lattice is Gelfand if and only if the hull–kernel and the -topology coincide on the set of maximal filters.
... Thus, for example, the definitions coincide for purity of ideals of R. The notion of pure ideals has been extensively studied during past decades, see e.g., Bkouche (1970), Fieldhouse (1974), De Marco (1983), Borceux and VanDenBossche (1983), Borceux et al. (1984), Al-Ezeh et al. (1988), Al-Ezeh (1989), Tarizadeh (2019), Tarizadeh and Aghajani (2021), Tarizadeh (2021). Also, pure ideals are investigated under the name of neat ideals in Johnstone (1982, p. 188). ...
This paper studies a fascinating type of filter in residuated lattices, the so-called pure filters. A combination of algebraic and topological methods on the pure filters of a residuated lattice is applied to obtain some new structural results. The notion of purely-prime filters of a residuated lattice has been investigated, and a Cohen-type theorem has been obtained. It is shown that the pure spectrum of a residuated lattice is a compact sober space, and a Grothendieck-type theorem has been demonstrated. It is proved that the pure spectrum of a Gelfand residuated lattice is a Hausdorff space, and deduced that the pure spectrum of a Gelfand residuated lattice is homeomorphic to its usual maximal spectrum. Finally, the pure spectrum of an mp-residuated lattice is investigated and verified that a given residuated lattice is mp iff its minimal prime spectrum is equipped with the induced dual hull-kernel topology, and its pure spectrum is the same.
... Gelfand rings have been the main subject of many articles in the literature over the years and are still of current interest, see e.g. Johnstone (1982); Contessa (1982Contessa ( , 1984; Al-Ezeh (1989, 1990a; Banaschewski (2000); Aghajani and Tarizadeh (2020). Motivated by De Marco and Orsatti (1971) the notion of a pm-lattice was introduced by Pawar and Thakare (1977) as a bounded distributive lattice in which any prime ideal is contained in a unique maximal ideal. ...
In this paper, a combination of algebraic and topological methods is applied to obtain new and structural results on Gelfand residuated lattices. It is demonstrated that Gelfand's residuated lattices strongly tied up with the hull-kernel topology. Especially, it is shown that a residuated lattice is Gelfand if and only if its prime spectrum, equipped with the hull-kernel topology, is normal. The class of soft residuated lattices is introduced, and it is shown that a residuated lattice is soft if and only if it is Gelfand and semisimple. Gelfand residuated lattices are characterized using the pure part of filters. The relation between pure filters and radicals in a Gelfand residuated lattice is described. It is shown that a residuated lattice is Gelfand if and only if its pure spectrum is homeomorphic to its usual maximal spectrum. The pure filters of a Gelfand residuated lattice are characterized. Finally, it is proved that a residuated lattice is Gelfand if and only if the hull-kernel and the -topology coincide on the set of maximal filters.
... Pure notion was studied in some works, e.g. [1], [2], [3], [4] and [6]. In §2 we study the notion of N-pure ideal. ...
In this paper, we introduce the concept of N-pure ideal as a generalization of pure ideal. Using this concept, a new and interesting type of rings is presented, we call it mid ring. Also, we provide new characterizations for von Neumann regular and zero-dimensional rings. Moreover, some results about mp-ring are given. Finally, a characterization for mid rings is provided. Then it is shown that the class of mid rings is strictly between the class of reduced mp-rings (p.f. rings) and the class of mp-rings.
... Following [1],an ideal I of a ring R is said to be right(left) pure ideal, if for any a I , there exists b I such that a = ab . (a = ba) . ...
As a generalization of right pure ideals, we introduce the notion of right П – pure ideals. A right ideal I of R is said to be П – pure, if for every a Î I there exists b Î I and a positive integer n such that an ≠ 0 and an b = an. In this paper, we give some characterizations and properties of П – pure ideals and it is proved that:
If every principal right ideal of a ring R is П – pure then,
a).L (an) = L (an+1) for every a Î R and for some positive integer n .
b). R is directly finite ring.
c). R is strongly П – regular ring.
... This concept was introduced by Fieldhouse [6] and study by AL-Ezeh [1], [2], [3]. As a generalization of this concept Shuker and Mahmood [11] defined right (left) GP-(generalized pure) ideals that is an ideal I of a ring R such that for every I a there exists I b and a positive integer n such that b a a n n = ) ( n n ba a = . ...
In this work we give some new properties of GP- ideals as well as the relation between GP- ideals, - π regular and simple ring. Also we consider rings with every principal ideal are GP- ideals and establish relation between such rings with strongly regular and local rings.
... Purely-prime notion was introduced and studied in [3,Chaps. 7 ,8] for general rings (not necessarily commutative), it is also studied in [1], [2]. However, except in these papers, this topic seem to have not been made the subject of special study. ...
In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring (pure spectrum) are obtained. Especially, Grothendieck type theorem is obtained which states that there is a canonical correspondence between the idempotents of a ring and the clopens of its pure spectrum. It is also proved that a given ring is a Gelfand ring iff its maximal spectrum equipped with the induced Zariski topology is homeomorphic to its pure spectrum. Then as an application, it is deduced that a ring is zero dimensional iff its prime spectrum and pure spectrum are isomorphic. Dually, it is shown that a given ring is a reduced mp-ring iff its minimal spectrum equipped with the induced flat topology and its pure spectrum are the same. Finally, the new notion of semi-Noetherian ring is introduced and Cohen type theorem is proved.
... Pure ideals are interesting because they classify important families of rings, e.g. Von Neumman rings and P F -rings, see Al-Ezeh [2,3,4,5], Jondrup [12], DeMarco [10]. Also, a ring R ∈ C , if for any pure ideal I in R, I is generated by a family of idempotents. ...
For a commutative ring with unity R, a commutative ring is defined called the ring of Hurwitz series, HR. As a subring of this ring, HR , the ring of Hurwitz polynomnials is defined, hR. In this paper, we characterize pure ideals in the ring hR. Then we characterize when the ring hR is an almost PP-ring and a PF-ring. Finally, for the ring R satisfying for a fixed positive integer n, we prove that every prime ideal of the rings HR(hR) is maximal and so their spectrum is completely characterized.
... We write for any ∈ , ( ) and ( ) the right annihilater of and the left annihilater of respectively . Standard references like [1] , [2] , and [6] have motivated many authors for study pure ideals . An ideal of a ring is called right pure if for every ∈ there exists ∈ such that = . ...
Let R be a ring. The ring R is called right Nil-pure, if for any í µí² ∈ í µí±¹ , r(a) is an aleft pure ideal of R. In this paper, we give some characterizations and properties of right Nil-pure rings, which is a proper generalization of every ideal of R is pure. And we study the regularity of right Nil-pure ring. For example: 1-Let R be a reversible and Nil-pure ring. Then R is n-regular ring 2-Let R be ZI-ring with every simple singular right R-module is almost nil-injective, then R is nil-pure ring
... An ideal I of a ring R is said to be right (left)pure if for every I a ∈ , there exists I b ∈ such that a=ab (a=ba), [1], [2]. Throughout this paper, R is an associative ring with unity. ...
MEP) ﺍﻟ ﻤﺜـﺎﻟﻲ ﺠﺯﺀ ﻜل ﻓﻴﻬﺎ ﺍﻟﺘﻲ ﺤﻠﻘﺎﺕ ﺍﻴﻤﻥ ﺃﻋﻅﻤﻲ ﺃﺴﺎﺴﻲ ﻨﻘﻲ ﻫﻭ ﺃﻴﺴﺭ (ﻭﺇﻋﻁﺎﺀ ﻟﻬـﺎ ﺍﻷﺴﺎﺴﻴﺔ ﺍﻟﺨﻭﺍﺹ. ﺍﻟـﺸﺭﻭﻁ ﺇﻋﻁـﺎﺀ ﻜـﺫﻟﻙ ﻟﻠﺤﻠﻘﺔ ﻭﺍﻟﻜﺎﻓﻴﺔ ﺍﻟﻀﺭﻭﺭﻴﺔ MEP ﺒﻀﻌﻑ ﻭﻤﻨﺘﻅﻤﺔ ﺒﻘﻭﺓ ﻤﻨﺘﻅﻤﺔ ﺤﻠﻘﺔ ﺘﻜﻭﻥ ﻟﻜﻲ. ABSTRACT This paper introduces the notion of a right MEP-ring (a ring in which every maximal essential right ideal is left pure) with some of their basic properties; we also give necessary and sufficient conditions for MEP-rings to be strongly regular rings and weakly regular rings.
... An ideal I of a ring R is said to be right(left) pure if for every a∈I, there exists b∈I such that a=ab (a=ba). This concept was introduced by Fieldhouse [6], [ 7 ], Al-Ezeh [ 2 ], [ 3 ] and Mahmood [ 9 ]. Recall that:-1-A ring R is regular if for every a∈R there exists b∈R such that a=aba, if a=a 2 b, R is called strongly regular. ...
The purpose of this paper is to study the class of the rings for which every maximal right ideal is left GP-ideal. Such rings are called MGP-rings and give some of their basic properties as well as the relation between
MGP-rings, strongly regular ring, weakly regular ring and kasch ring.
... central. [6] 2-PILP-Rings: Following [1], an ideal I of a ring R is said to be a left (right) pure if for every a ∈ I, there exists b ∈ I such that a = ba(a=ab). ...
In this paper, we study rings whose principal right ideals are left pure. Also we shall introduce the concept of a fuzzy bi-ideal in a ring, and give some properties of such fuzzy ideals. We also give a characterization of whose principal right ideal are left pure, fuzzy duo ring in terms of fuzzy deals.
... Following [1], an ideal I is said to be a right (left) pure if for every ...
In this work, we study rings whose every principal ideal is a right pure. We give some properties of right PIP – rings and the connection between such rings and division rings.
This paper is devoted to the study of a fascinating class of residuated lattices, the so-called mp-residuated lattice, in which any prime filter contains a unique minimal prime filter. A combination of algebraic and topological methods is applied to obtain new and structural results on mp-residuated lattices. It is demonstrated that mp-residuated lattices are strongly tied up with the dual hull-kernel topology. Especially, it is shown that a residuated lattice is mp if and only if its minimal prime spectrum, equipped with the dual hull-kernel topology, is Hausdorff if and only if its prime spectrum, equipped with the dual hull-kernel topology, is normal. The class of mp-residuated lattices is characterized by means of pure filters. It is shown that a residuated lattice is mp if and only if its pure filters are precisely its minimal prime filters, if and only if its pure spectrum is homeomorphic to its minimal prime spectrum, equipped with the dual hull-kernel topology.
A subalgebra A(X) of C(X) is said to be a \beta-subalgebra if the space of maximal ideals of A(X) equipped with the hull-kernel topology is homeomorphic to \beta X. Kharbhih and Dutta in [Closure formula for ideals in intermediate rings, Appl. Gen. Topol. 21 (2) (2020), 195-200] showed that the closure of every ideal I of an intermediate ring under the m-topology, briefly, the m-closure of I, equals the intersection of all maximal ideals in A(X) containing I. In this paper, we extend this fact to the class of \beta-subalgebras which is shown to be a larger class than intermediate rings. We also study a more extended class of subrings than
\beta-subalgebras, namely, LBI-subalgebras, and characterize the conditions under which an LBI-subalgebra is a \beta-subalgebra. Moreover, some known facts in the context of C(X) and intermediate rings of C(X) are generalized to \beta-subalgebras.
We return to the work of Banaschewski and extract from it a theorem of Fuchs, Heinzer, and Olberding. As an application of Fuchs-Heinzer-Olberding's theorem, we generalize a result of Gillman and Kohls. We study pseudo-irreducible ideals and show that every ideal of a pm-ring is the (not necessarily finite) intersection of pairwise comaximal pseudo-irreducible ideals. After some general results, the article focuses on primal and pseudo-irreducible ideals in rings of continuous functions. We determine when every pseudo-irreducible ideal of C(X) is primal. We give a characterization of spaces X for which every is a primal ideal of C(X).
This paper is devoted to the study of a fascinating class of residuated lattices, the so-called mp-residuated lattice, in which any prime filter contains a unique minimal prime filter. A combination of algebraic and topological methods is applied to obtain new and structural results on mp-residuated lattices. It is demonstrated that mp-residuated lattices are strongly tied up with the dual hull-kernel topology. Especially, it is shown that a residuated lattice is mp if and only if its minimal prime spectrum, equipped with the dual hull-kernel topology, is Hausdorff if and only if its prime spectrum, equipped with the dual hull-kernel topology, is normal. The class of mp-residuated lattices is characterized by means of pure filters. It is shown that a residuated lattice is mp if and only if its pure filters are precisely its minimal prime filters, if and only if its pure spectrum is homeomorphic to its minimal prime spectrum, equipped with the dual hull-kernel topology.
In this paper, the notion of a Rickart residuated lattice is introduced and investigated. A residuated lattice is called Rickart if any its coannulet is generated by a complemented element. It is observed that a residuated lattice , is Rickart iff any its coannulet is a direct summand of iff it is quasicomplemented and normal iff it is generalized Stone. Some algebraic and topological characterizations are obtained, and some facts about pure and Stone filters are also extracted, which are given in the paper.
In this paper, new criteria for zero dimensional rings, Gelfand rings, clean rings and mp-rings are given. A new class of rings is introduced and studied, we call them purified rings. Specially, reduced purified rings are characterized. New characterizations for pure ideals of reduced Gelfand rings and mp-rings are provided. It is also proved that if the topology of a scheme is Hausdorff, then the affine opens of that scheme is stable under taking finite unions. In particular, every compact scheme is an affine scheme.
The aim of this paper is twofold. Firstly, to give short proofs of the celebrated results of Rudd (1970), De Marco (1972), Brookshear (1977), Neville (1990), and Vechtomov (1996) with the help of some well-known results in ring theory. Secondly, to establish new algebraic characterizations for when C(X) is von Neumann regular, semihereditary, or a B'ezout ring. The notions of greatest common divisor and least common multiple in the ring C(X) are studied. Finally, the spaces X for which every idempotent of the classical quotient ring of C(X) is actually an element of C(X) itself, is completely determined.
An ideal I of a ring R is said to be right (left) Pure if for every , there is such that . A ring R is said to be right (left) MP-ring, if every maximal right (left) ideal of R is a left (right) pure. In this paper have been studied some new properties of MP-rings, there connections with strongly regular rings.
Some of the main result of the present work are as follows:
1- Let R be aright MP-ring, r(a) is a W-ideal for all then
a- Every essential ideal is a direct summand.
b- R is strongly regular ring.
2- Let R be aright MP-ring. If R is right almost abelian left NBF ring, then R is strongly regular.
All rings considered are commutative with unity and all groups considered are abelian. We give a characterization of a pure augmentation ideal, I(G), of a group ring, R(G). We study the relationship between the p-injectivity of R(G) and the p-injectivity of its ideal I(G). Keywords and phrases: Augmentation ideal, Pure ideal, P-injective ring, P-injective ideal.
All rings R considered are commutative and have an identity element. Contessa called R a VNL-ring if a or 1 - a has a Von Neumann inverse whenever a ∈ R. Sample results: Every prime ideal of a VNL-ring is contained in a unique maximal ideal. Local and Von Neumann regular rings are VNL and if the product of two rings is VNL, then both are Von Neumann regular, or one is Von Neumann regular and the other is VNL. The ring n of integers mod n is VNL iff (pq)2 ∤ n whenever p and q are distinct primes. The ring R[[x]] of formal power series over R is VNL iff R is local. The ring C(X) of all continuous real-valued functions on a Tychonoff space X is VNL if and only if at most one point of X fails to be a P-point. All known VNL-rings satisfy SVNL, namely whenever the ideal generated by a (finite) subset of R is all of R, one of its members has a Von Neumann inverse. We show that a ring R is SVNL if and only if all maximal ideals of R are pure except maybe one. We show that ∏∈IR() is an SVNL if and only if there exists 0 ∈ I, such that R(0) is an SVNL and for all ∈ I - 0, R() is a Von Neumann regular ring. Whether every VNL-ring is an SVNL is an open question.
For a commutative ring with unity R, it is proved that R is a PF-ring if and only if the annihilator, annR(a), for each a õ R is a pure ideal in R, Also it is proved that the polynomial ring, R[X], is a PF-ring if and only if R is a PF-ring. Finally, we prove that R is a PP-ring if and only if R[X] is a PP-ring.
Categories of algebraic sheaves.- The formal initial segments.- Localizations and algebraic sheaves.- Integral theories and characterization theorem.- Spectrum of a theory.- Applications to module theory.- Pure representation of rings.- Gelfand rings.
Rings of continuous functions. Graduate Texts in Math
- L Gillman
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On PM-rings. Comm. Algebra10
- M. Contessa