We study the central reflexive properties of rings with an involution. The concept of central *-reflexive rings is introduced and investigated. It is shown that central *-reflexive rings are a generalization of reflex...We study the central reflexive properties of rings with an involution. The concept of central *-reflexive rings is introduced and investigated. It is shown that central *-reflexive rings are a generalization of reflexive rings, central reflexive rings and *-reflexive rings. Some characterizations of this class of rings are given. The related ring extensions including trivial extension, Dorroh extension and polynomial extensions are also studied.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
A ring R is said to be quasi-central semicommutative(simply,a QCS ring)if ab=0 implies aRb⊆Q(R)for a,b∈R,where Q(R)is the quasi-center of R.It is proved that if R is a QCS ring,then the set of nilpotent elements of R...A ring R is said to be quasi-central semicommutative(simply,a QCS ring)if ab=0 implies aRb⊆Q(R)for a,b∈R,where Q(R)is the quasi-center of R.It is proved that if R is a QCS ring,then the set of nilpotent elements of R coincides with its Wedderburn radical,and that the upper triangular matrix ring R=Tn(S)for any n≥2 is a QCS ring if and only if n=2 and S is a duo ring,while T k 2k+2(R)is a QCS ring when R is a reduced duo ring.展开更多
Let G be a finite group, H ≤ G and R be a commutative ring with an identity 1R. Let CRG（H）={α ∈ RG｜αh = hα for all h ∈ H）, which is called the centralizer subalgebra of H in RG. Obviously, if H=G then CRG（H...Let G be a finite group, H ≤ G and R be a commutative ring with an identity 1R. Let CRG（H）={α ∈ RG｜αh = hα for all h ∈ H）, which is called the centralizer subalgebra of H in RG. Obviously, if H=G then CRG（H） is just the central subalgebra Z（RG） of RG. In this note, we show that the set of all H- conjugacy class sums of G forms an R-basis of CRG（H）. Furthermore, let N be a normal subgroup of G and γthe natural epimorphism from G to G to G/N. Then γ induces an epimorphism from RG to RG, also denoted by % We also show that if R is a field of characteristic zero, then γ induces an epimorphism from CRG（H） to CRG（H）, that is, 7（CRG（H）） = CRG（H）.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11601005)。
文摘We study the central reflexive properties of rings with an involution. The concept of central *-reflexive rings is introduced and investigated. It is shown that central *-reflexive rings are a generalization of reflexive rings, central reflexive rings and *-reflexive rings. Some characterizations of this class of rings are given. The related ring extensions including trivial extension, Dorroh extension and polynomial extensions are also studied.
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.
基金Supported by the National Nature Science Foundation of China(Grant No.61972235).
文摘A ring R is said to be quasi-central semicommutative(simply,a QCS ring)if ab=0 implies aRb⊆Q(R)for a,b∈R,where Q(R)is the quasi-center of R.It is proved that if R is a QCS ring,then the set of nilpotent elements of R coincides with its Wedderburn radical,and that the upper triangular matrix ring R=Tn(S)for any n≥2 is a QCS ring if and only if n=2 and S is a duo ring,while T k 2k+2(R)is a QCS ring when R is a reduced duo ring.
基金The NSF(11071155) of Chinathe NSF(2008A03) of Shandong Province
文摘Let G be a finite group, H ≤ G and R be a commutative ring with an identity 1R. Let CRG（H）={α ∈ RG｜αh = hα for all h ∈ H）, which is called the centralizer subalgebra of H in RG. Obviously, if H=G then CRG（H） is just the central subalgebra Z（RG） of RG. In this note, we show that the set of all H- conjugacy class sums of G forms an R-basis of CRG（H）. Furthermore, let N be a normal subgroup of G and γthe natural epimorphism from G to G to G/N. Then γ induces an epimorphism from RG to RG, also denoted by % We also show that if R is a field of characteristic zero, then γ induces an epimorphism from CRG（H） to CRG（H）, that is, 7（CRG（H）） = CRG（H）.
