In this paper,potent index of an element and pseudo clean rings are considered.Some properties and examples of pseudo clean rings are given.We also show that Zm is pseudo clean for every 2≤m∈Z and pseudo clean rings...In this paper,potent index of an element and pseudo clean rings are considered.Some properties and examples of pseudo clean rings are given.We also show that Zm is pseudo clean for every 2≤m∈Z and pseudo clean rings are clean.Furthermore,we prove pseudo clean rings are directly finite and have stable range one.展开更多
As generalization of r-clean rings and weakly clean rings, we define a ring R is weakly r-clean if for any a E R there exist an idempotent e and a regular element r such that a = r ＋ e or a = r - e. Some properties a...As generalization of r-clean rings and weakly clean rings, we define a ring R is weakly r-clean if for any a E R there exist an idempotent e and a regular element r such that a = r ＋ e or a = r - e. Some properties and examples of weakly r-clean rings are given. Furthermore, we prove the weakly clean and weakly r-clean rings are equivalent for abelian rings.展开更多
A ＊-ring is called ＊-clean if every element of the ring can be written as the sum of a projection and a unit. For an integer n ≥ 1, we call a ＊-ring R n-＊-clean if for any a ∈ R,a = p ＋ u1 ＋ ···...A ＊-ring is called ＊-clean if every element of the ring can be written as the sum of a projection and a unit. For an integer n ≥ 1, we call a ＊-ring R n-＊-clean if for any a ∈ R,a = p ＋ u1 ＋ ··· ＋ unwhere p is a projection and ui are units for all i. Basic properties of n-＊-clean rings are considered, and a number of illustrative examples of 2-＊-clean rings which are not ＊-clean are provided. In addition, extension properties of n-＊-clean rings are discussed.展开更多
We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of...We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring Mn（R） is weakly nil-clean, and to show that the endomorphism ring EndD（V） over a vector space VD is weakly nil-clean if and only if it is nil-clean or dim（V） = 1 with D Z3.展开更多
A ring R with unity is called semiclean, if each of its elements is the sum of a unit and a periodic. Every clean ring is semiclean. It is not easy to characterize a semiclean group ring in general. Our purpose is to ...A ring R with unity is called semiclean, if each of its elements is the sum of a unit and a periodic. Every clean ring is semiclean. It is not easy to characterize a semiclean group ring in general. Our purpose is to consider the following question： If G is a locally finite group or a cyclic group of order 3, then when is RG semiclean？ Some known results on clean group rings are generalized.展开更多
An element a of a ring R is called uniquely strongly clean if it is the sum of an idempotent and a unit that commute, and in addition, this expression is unique. R is called uniquely strongly clean if every element of...An element a of a ring R is called uniquely strongly clean if it is the sum of an idempotent and a unit that commute, and in addition, this expression is unique. R is called uniquely strongly clean if every element of R is uniquely strongly clean. The uniquely strong cleanness of the triangular matrix ring is studied. Let R be a local ring. It is shown that any n × n upper triangular matrix ring over R is uniquely strongly clean if and only if R is uniquely bleached and R/J（R） ≈Z2.展开更多
A ring R is called clean if every element is the sum of an idempotent and a unit, and R is called uniquely strongly clean （USC for short） if every element is uniquely the sum of an idempotent and a unit that commute...A ring R is called clean if every element is the sum of an idempotent and a unit, and R is called uniquely strongly clean （USC for short） if every element is uniquely the sum of an idempotent and a unit that commute. In this article, some conditions on a ring R and a group G such that RG is clean are given. It is also shown that if G is a locally finite group, then the group ring RG is USC if and only if R is USC, and G is a 2-group. The left uniquely exchange group ring, as a middle ring of the uniquely clean ring and the USC ring, does not possess this property, and so does the uniquely exchange group ring.展开更多
Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u.Moreover,a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element o...Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u.Moreover,a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element of R is centrally clean.First,some characterizations of centrally clean elements are given.Furthermore,some properties of centrally clean rings,as well as the necessary and sufficient conditions for R to be a centrally clean ring are investigated.Centrally clean rings are closely related to the central Drazin inverses.Then,in terms of centrally clean decomposition,the necessary and sufficient conditions for the existence of central Drazin inverses are presented.Moreover,the central cleanness of special rings,such as corner rings,the ring of formal power series over ring R,and a direct product ∏ R_(α) of ring R_(α),is analyzed.Furthermore,the central group invertibility of combinations of two central idempotents in the algebra over a field is investigated.Finally,as an application,an example that lists all invertible,central group invertible,group invertible,central Drazin invertible elements,and centrally clean elements of the group ring Z_(2)S_(3) is given.展开更多
基金Supported by National Natural Science Foundation of China(12301041)。
文摘In this paper,potent index of an element and pseudo clean rings are considered.Some properties and examples of pseudo clean rings are given.We also show that Zm is pseudo clean for every 2≤m∈Z and pseudo clean rings are clean.Furthermore,we prove pseudo clean rings are directly finite and have stable range one.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1140100911326062)
文摘As generalization of r-clean rings and weakly clean rings, we define a ring R is weakly r-clean if for any a E R there exist an idempotent e and a regular element r such that a = r ＋ e or a = r - e. Some properties and examples of weakly r-clean rings are given. Furthermore, we prove the weakly clean and weakly r-clean rings are equivalent for abelian rings.
基金Supported by the National Natural Science Foundation of China(Grant No.11401009)the Natural Science Foundation of Anhui Province(Grant No.1408085QA01)+1 种基金the Key Natural Science Foundation of Anhui Educational Committee(Grant No.KJ2014A082)the Educational Foundation of Anhui Normal University(Grant No.2013qnzx35)
文摘A ＊-ring is called ＊-clean if every element of the ring can be written as the sum of a projection and a unit. For an integer n ≥ 1, we call a ＊-ring R n-＊-clean if for any a ∈ R,a = p ＋ u1 ＋ ··· ＋ unwhere p is a projection and ui are units for all i. Basic properties of n-＊-clean rings are considered, and a number of illustrative examples of 2-＊-clean rings which are not ＊-clean are provided. In addition, extension properties of n-＊-clean rings are discussed.
文摘We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring Mn（R） is weakly nil-clean, and to show that the endomorphism ring EndD（V） over a vector space VD is weakly nil-clean if and only if it is nil-clean or dim（V） = 1 with D Z3.
基金Supported by the National Natural Science Foundation of China(Grant No.11401009)the Natural Science Foundation of Ahui Province(Grant No.1408085QA01)
文摘A ring R with unity is called semiclean, if each of its elements is the sum of a unit and a periodic. Every clean ring is semiclean. It is not easy to characterize a semiclean group ring in general. Our purpose is to consider the following question： If G is a locally finite group or a cyclic group of order 3, then when is RG semiclean？ Some known results on clean group rings are generalized.
基金The National Natural Science Foundation of China(No.10971024)the Specialized Research Fund for the Doctoral Program of Higher Education(No.200802860024)the Natural Science Foundation of Jiangsu Province(No.BK2010393)
文摘An element a of a ring R is called uniquely strongly clean if it is the sum of an idempotent and a unit that commute, and in addition, this expression is unique. R is called uniquely strongly clean if every element of R is uniquely strongly clean. The uniquely strong cleanness of the triangular matrix ring is studied. Let R be a local ring. It is shown that any n × n upper triangular matrix ring over R is uniquely strongly clean if and only if R is uniquely bleached and R/J（R） ≈Z2.
文摘A ring R is called clean if every element is the sum of an idempotent and a unit, and R is called uniquely strongly clean （USC for short） if every element is uniquely the sum of an idempotent and a unit that commute. In this article, some conditions on a ring R and a group G such that RG is clean are given. It is also shown that if G is a locally finite group, then the group ring RG is USC if and only if R is USC, and G is a 2-group. The left uniquely exchange group ring, as a middle ring of the uniquely clean ring and the USC ring, does not possess this property, and so does the uniquely exchange group ring.
基金The National Natural Science Foundation of China(No.12171083,11871145,12071070)the Qing Lan Project of Jiangsu Province。
文摘Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u.Moreover,a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element of R is centrally clean.First,some characterizations of centrally clean elements are given.Furthermore,some properties of centrally clean rings,as well as the necessary and sufficient conditions for R to be a centrally clean ring are investigated.Centrally clean rings are closely related to the central Drazin inverses.Then,in terms of centrally clean decomposition,the necessary and sufficient conditions for the existence of central Drazin inverses are presented.Moreover,the central cleanness of special rings,such as corner rings,the ring of formal power series over ring R,and a direct product ∏ R_(α) of ring R_(α),is analyzed.Furthermore,the central group invertibility of combinations of two central idempotents in the algebra over a field is investigated.Finally,as an application,an example that lists all invertible,central group invertible,group invertible,central Drazin invertible elements,and centrally clean elements of the group ring Z_(2)S_(3) is given.
