No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
On generalized PF-rings
... Al-Ezeh in [4] introduced a new class of rings called GP F −rings. A ring R is called a GP F −ring if for each a ∈ R, there exists a positive integer n such that a n R is a flat R−module. ...
... Also, a ring R is a GP F −ring if and only if for every a ∈ R, there exists a positive integer n such that the annihilator ideal Ann R (a n ) is a pure ideal. A study of this class of rings and the relationship between GP P −rings and GP F −rings were carried by Al-Ezeh [4]. ...
... Every P F −ring is a reduced ring (a ring with no nonzero nilpotent elements), see [2, Lemma 2], but not every GP F −ring is a reduced ring. Al-Ezeh in [4] proved that a ring R is a reduced GP F −ring if and only if R is a P F −ring. Let C(X) be the ring of all real valued continuous functions defined on X. ...
All rings R in this article are assumed to be commutative with unity 1 = 0. A ring R is called a GP F −ring if for every a ∈ R there exists a positive integer n such that the annihilator ideal Ann R (a n) is pure. We prove that for a ring R and an Abelian group G, if the group ring RG is a GP F −ring then so is R. Moreover, if G is a finite Abelian group then |G| is a unit or a zero-divisor in R. We prove that if G is a group such that for every nontrivial subgroup H of G, [G : H] < ∞, then the group ring RG is a GP F −ring if and only if RH is a GP F −ring for each finitely generated subgroup H of G. It is proved that if R is a local ring and RG is a U −group ring, then RG is a GP F −ring if and only if R is a GP F −ring and p ∈ N il (R). Finally, we prove that if R is a semisimple ring and G is a finite group such that |G| −1 ∈ R, then RG is a GP F −ring if and only if RG is a P F −ring.
... It is proved in [1] that a ring R is a PF-ring if and only if the annihilator of each element r ∈ R, ann R (r), is a pure ideal; that is, for all b ∈ ann R (r) there exists c ∈ ann R (r) such that bc = b. It may be worth reminding the reader that for a commutative ring R, R is a PF-ring if and only if R is a locally integral domain (i.e., every localization R P is an integral domain for any prime (resp., maximal) ideal P of R) ( [3], [11]). It is also proved in [2] that the power series ring R[ [X]] is a PF-ring if and only if for any two countable subsets A = {a 0 , a 1 , · · · } and B = {b 0 , b 1 , · · · } of R such that A ⊆ ann R (B), there exists r ∈ ann R (B) such that ar = a for all a ∈ A. In [7, Theorem 3, Theorem 4], J.-H. ...
... Recall that R called a generalized PF-ring (for short, GPF-ring) if, given any a ∈ R, then the principal ideal Ra n is flat as an R-module for some n ≥ 1. It is proved in [3] that a commutative ring R is a GPF-ring if and only if, given any a ∈ R, either a is a regular element in every prime localization R P or for some n ≥ 1, a n = 0 in every R P . Also note in [3] that a commutative ring R is a PF-ring if and only if R is a reduced GPF-ring. ...
... It is proved in [3] that a commutative ring R is a GPF-ring if and only if, given any a ∈ R, either a is a regular element in every prime localization R P or for some n ≥ 1, a n = 0 in every R P . Also note in [3] that a commutative ring R is a PF-ring if and only if R is a reduced GPF-ring. ...
In this paper, we show that if R is a commutative ring with identity and (S, ) is a strictly totally ordered monoid, then the ring [[]] of generalized power series is a PF-ring if and only if for any two S-indexed subsets A and B of R such that , there exists such that for all , and that for a Noetherian ring R, ]] is a PP ring if and only if R is a PP ring.
... Endo [5] and Hirano [7] generalized p.p. ring notion by defining that a ring R is said to be a generalized p.p. ring (or, GPP-ring) if for each f ∈ R there exists a natural number n ≥ 1 such that Rf n is R−projective. Motivated by the Hirano's work then in [2], the generalized p.f. ring notion is also defined which states that a ring R is called a generalized p.f. ring (or, GPF-ring) if for each f ∈ R there exists some n ≥ 1 such that Rf n is a flat R−module. Every GPP-ring is a GPF-ring. ...
... In the same vein, we also prove that a ring R is a p.f. ring if and only if the polynomial ring R[x] is a p.f. ring, see Theorem 2.5. We have also made some significant improvements in the main results of [2] and [7] which are including Theorems 2.6, 2.13, 2.16, 3.3 and Corollary 3.5. Under a mild condition, it is shown that a ring is a generalized p.f. ring if and only if it is locally primary ring, see Theorems 3.10 and 3.2. ...
In this paper, new and significant advances on the structure of p.p. rings and their generalizations have been made. Especially amongst them, it is proved that a commutative ring R is a generalized p.p. ring if and only if R is a generalized p.f. ring and its minimal spectrum is Zariski compact. Some of the major results of the literature related to this context either are improved or are proven by new methods. Specially, we give a new and elementary proof to the fact that a commutative ring R is a p.p. ring if and only if R[x] is a p.p. ring. We also prove the same assertion for p.f. rings. Finally, the new notion of soft ring is introduced and studied which generalizes at once both generalized p.p. ring and almost p.p. ring notions.
... 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.
... 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.
... Therefore Im n f is a direct summand of R, and hence R dual -Rickart, in other words, R is a -regular rings. ann a isa pure ideal of R [2], in other words, R is Pure -Rickart. The following proposition shows that the dual Pure -Rickart property is inherited from a module to each of its direct summands. ...
Let R be a commutative ring with identity and M be an R-module.In this paper we introducethe dual concepts of Pure Rickart modules and Pureí µí¼-Rickart modulesas a generalization of dual Rickart modules anddual í µí¼-Rickart modules respectively. Further, dual Pure Rickart modules and dual Pure í µí¼-Rickart modules can be consideredas a generalization of regular rings and í µí¼-regular rings respectively. Furthermore, dual Pureí µí¼-Rickart modulesis a generalization of PureRickart modules. An R-module M is calleddualPure Rickartif for every f∈EndR(M),Im f is a pure(in sense of Anderson and Fuller) submodule of M. An R-module M is calleddual Pure í µí¼-Rickart if for every f∈EndR(M), there exist a positive integer n such that (M), there exist a positive integer n such that is a pure (in sense of Anderson and Fuller) submodule of M.We show that several results of dual Rickart modules anddual í µí¼-Rickart modules can be extended to dual Pure Rickart modules and dual Pureí µí¼-Rickart modules for this general settings. Many results about these concepts are givenand some relationships between these modules and other related modules are investigated.
... Therefore when M = R, the concept of Pure í µí¼-Rickart modules coincides with that of GPF-rings. A ring R is called GPF-ring if for every a ∈ R, there exists a positive integer n such that () n R ann a is a pure ideal of R [2]. ...
... Let HR be an ID and x = (0, 1, 0, 0, ...) ∈ HR. Then(1) x is irreducible element.(2) x is prime element if and only if for all n ∈ {1, 2, ...}, n ∈ U(R) .Proof. ...
In this paper, new advances on the understanding the structure of p.p. rings and their generalizations have been made. Especially among them, it is proved that a commutative ring R is a generalized p.p. ring if and only if R is a generalized p.f. ring and its minimal spectrum is Zariski compact, or equivalently, is a p.p. ring and is a primary ring for all . Some of the major results of the literature either are improved or are proven by new methods. In particular, we give a new and quite elementary proof to the fact that a commutative ring R is a p.p. ring if and only if R[x] is a p.p. ring.
Let R be a ring which is S-compatible and (S,ω)-Armendariz. In this paper, we investigate that the skew generalized power series ring R[[S,ω]] is a PF-ring if and only if for any two S-indexed subsets P and Q of R such that Q⊆ann R (P) and there exists a∈ann R (P) such that qa=q for all q∈Q. Further, we prove that if R be a Noetherian ring then R[[S,ω]] is a PP-ring if and only if R is a PP-ring.
ResearchGate has not been able to resolve any references for this publication.