We introduce the concepts of left （right） zero-divisor rings, a class of rings without identity. We call a ring R left （right） zero-divisor if rR（a） ≠ 0（lR（a） ≠ 0） for every a∈ R, and call R strong left ...We introduce the concepts of left （right） zero-divisor rings, a class of rings without identity. We call a ring R left （right） zero-divisor if rR（a） ≠ 0（lR（a） ≠ 0） for every a∈ R, and call R strong left （right） zero-divisor if r R （R）≠0（lR（R）≠ 0）. Camillo and Nielson called a ring right finite annihilated （RFA） if every finite subset has non-zero right annihilator. We present in this paper some basic examples of left zero-divisor rings, and investigate the extensions of strong left zero-divisor rings and RFA rings, giving their equivalent characterizations.展开更多
A ring R is said to be π-semicommutative if a, b C R satisfy ab = 0 then there exists a positive integer n such that a n Rb n = O. We study the properties of π-semicommutative rings and the relationship between suc...A ring R is said to be π-semicommutative if a, b C R satisfy ab = 0 then there exists a positive integer n such that a n Rb n = O. We study the properties of π-semicommutative rings and the relationship between such rings and other related rings. In particular, we answer a question on left GWZI rings negatively.展开更多
Let us call a ring R (without identity) to be right symmetric if for any triple a,b,c,∈R?abc = 0 then acb = 0. Such rings are neither symmetric nor reversible (in general) but are semicommutative. With an idempotent ...Let us call a ring R (without identity) to be right symmetric if for any triple a,b,c,∈R?abc = 0 then acb = 0. Such rings are neither symmetric nor reversible (in general) but are semicommutative. With an idempotent they take care of the sheaf representation as obtained by Lambek. Klein 4-rings and their several generalizations and extensions are proved to be members of such class of rings. An extension obtained is a McCoy ring and its power series ring is also proved to be a McCoy ring.展开更多
Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized p...Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized power series reflexive and nil generalized power series reflexive, respectively. We obtain various necessary or sufficient conditions for a ring to be generalized power series reflexive and nil generalized power series reflexive. Examples are given to show that, nil generalized power series reflexive need not be generalized power series reflexive and vice versa, and nil generalized power series reflexive but not semicommutative are presented. We proved that, if R is a left APP-ring, then R is generalized power series reflexive, and R is nil generalized power series reflexive if and only if R/I is nil generalized power series reflexive. Moreover, we investigate ring extensions which have roles in ring theory.展开更多
Let R be a ring. An element a of R is called a left PP-element if Ra is projective. The ring R is said to be a left almost PP-ring provided that for any element a of R, either a or 1 - α is left PP. We develop, in th...Let R be a ring. An element a of R is called a left PP-element if Ra is projective. The ring R is said to be a left almost PP-ring provided that for any element a of R, either a or 1 - α is left PP. We develop, in this paper, left almost PP-rings as a generalization of von Neumann local （VNL） rings and left PP-rings. Some properties of left almost PP-rings are studied and some examples are also constructed.展开更多
In This paper, the concept of weakly dual ring is introduced, which is a proper generalization of the dual ring. If R is a right weakly dual ring, then (1) Z(RR) = J(R); (2) If R is also a zero-division power ring, th...In This paper, the concept of weakly dual ring is introduced, which is a proper generalization of the dual ring. If R is a right weakly dual ring, then (1) Z(RR) = J(R); (2) If R is also a zero-division power ring, then R is a right AP-injective ring. In addition, some properties of weakly dual rings are given.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.1107109711101217)
文摘We introduce the concepts of left （right） zero-divisor rings, a class of rings without identity. We call a ring R left （right） zero-divisor if rR（a） ≠ 0（lR（a） ≠ 0） for every a∈ R, and call R strong left （right） zero-divisor if r R （R）≠0（lR（R）≠ 0）. Camillo and Nielson called a ring right finite annihilated （RFA） if every finite subset has non-zero right annihilator. We present in this paper some basic examples of left zero-divisor rings, and investigate the extensions of strong left zero-divisor rings and RFA rings, giving their equivalent characterizations.
基金Supported by the Natural Foundation of Shandong Province(Grant Nos.ZR2013AL013ZR2014AL001)
文摘A ring R is said to be π-semicommutative if a, b C R satisfy ab = 0 then there exists a positive integer n such that a n Rb n = O. We study the properties of π-semicommutative rings and the relationship between such rings and other related rings. In particular, we answer a question on left GWZI rings negatively.
文摘Let us call a ring R (without identity) to be right symmetric if for any triple a,b,c,∈R?abc = 0 then acb = 0. Such rings are neither symmetric nor reversible (in general) but are semicommutative. With an idempotent they take care of the sheaf representation as obtained by Lambek. Klein 4-rings and their several generalizations and extensions are proved to be members of such class of rings. An extension obtained is a McCoy ring and its power series ring is also proved to be a McCoy ring.
文摘Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized power series reflexive and nil generalized power series reflexive, respectively. We obtain various necessary or sufficient conditions for a ring to be generalized power series reflexive and nil generalized power series reflexive. Examples are given to show that, nil generalized power series reflexive need not be generalized power series reflexive and vice versa, and nil generalized power series reflexive but not semicommutative are presented. We proved that, if R is a left APP-ring, then R is generalized power series reflexive, and R is nil generalized power series reflexive if and only if R/I is nil generalized power series reflexive. Moreover, we investigate ring extensions which have roles in ring theory.
基金Supported by the Natural Science Foundation of Hunan Province(Grant No.2016JJ2050)
文摘Let R be a ring. An element a of R is called a left PP-element if Ra is projective. The ring R is said to be a left almost PP-ring provided that for any element a of R, either a or 1 - α is left PP. We develop, in this paper, left almost PP-rings as a generalization of von Neumann local （VNL） rings and left PP-rings. Some properties of left almost PP-rings are studied and some examples are also constructed.
基金Foundationitem:The NNSP(19971073) of China and the NSF of Yangzhou University
文摘In This paper, the concept of weakly dual ring is introduced, which is a proper generalization of the dual ring. If R is a right weakly dual ring, then (1) Z(RR) = J(R); (2) If R is also a zero-division power ring, then R is a right AP-injective ring. In addition, some properties of weakly dual rings are given.
