Extensions of strongly π-regular general rings

The concept of the strongly π-regular general ring （with or without unity） is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- regular general rings are provided. It is shown that I is strongly π-regular if and only if, for each x ∈I, x^n =x^n＋1y = zx^n＋1 for n ≥ 1 and y, z ∈ I if and only if every element of I is strongly π-regular. It is also proved that every upper triangular matrix general ring over a strongly π-regular general ring is strongly π-regular and the trivial extension of the strongly π-regular general ring is strongly clean.

Uniquely strongly clean triangular matrix rings

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.

On Strongly Regular Rings

In this paper we investigate strongly regular rings. In terms of W-ideals of rings some characterizations of strongly regular rings are given.

Extensions of Morphic Rings

A ring R is called left morphic, if for any a ∈ R, there exists b ∈ R such that lR（a） =Rb and lR（b）= Ra. In this paper, we use the method which is different from that of Lee and Zhou to investigate when R[x, σ]/（x^n） is （left） morphic and when the ideal extension E（R, V） is （left） morphic. It is mainly shown that： （1） If is an automorphism of a division ring R, then S = R[x, σ]/（x^n） （n 〉 1） is a special ring. （2） If d,m are positive integers and n = dm, then E（Zn, mZn） is a morphic ring if and only if gcd（d, m） = 1.

I-semi-π-regular Rings

Let R be a ring and I an ideal of R. A ring R is called I-semi-π--regular if R/I is π-regular and idempotents of R can be strongly lifted modulo I. Characterizations of I-semi-π-regular rings are given and relations between semi-π-regular rings and semiregular rings are explored.

Strongly Clean Matrix Rings over a Skew Monoid Ring

Let R be a ring with an endomorphismσ,F∪{0}the free monoid generated by U={u1,…,ut}with 0 added,and M a factor of F obtained by setting certain monomials in F to 0 such that M^(n)=0 for some n.Then we can form the non-semiprime skew monoid ring R[M;σ].A local ring R is called bleached if for any j∈J(R)and any u∈U(R),the abelian group endomorphisms l_(u)−r_(j) and l_(j)−r_(u) of R are surjective.Using R[M;σ],we provide various classes of both bleached and non-bleached local rings.One of the main problems concerning strongly clean rings is to characterize the rings R for which the matrix ring M_(n)(R)is strongly clean.We investigate the strong cleanness of the full matrix rings over the skew monoid ring R[M;σ].

The Metapositive Definite Self－Conjugate Solution of the Matrix Equation AXB＝C over a Skew Field

Let F be the strong p-division ring [4]. This paper is sequel to [1]. Metapositive definite self-conjugate matrix over F is defined and the necessary and sufficient conditions for determining whether a partitioned matrix over F is metapositive definite self-conjugate are given.Moreover,a decomposition of pairwise matrices over F with the same numbers of columns is also presented. Whence some necessary and sufficient conditions for the existence of and the explicit expression for the metapositive definite self-conjugate solution of the matrix equation AXB=C over F are derived.

von Neumann正则环与右SF-环

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this paper, we study the regularity of right SF-rings and prove that if R is a right SF-ring whose all maximal (essential) right ideals are GW-ideals, then R is regular.

Reversible Properties of Monoid Crossed Products

We study the reversible properties of monoid crossed products. The new class of strongly CM-reversible rings is introduced and characterized. This class of rings is a generalization of those of strongly reversible rings, skew strongly reversible rings and strongly M-reversible rings. Some well-known results on this subject are generalized and extended.

Existence of Multiple Positive Periodic Solutions for Second Order Differential Equations

Let α be a nonzero endomorphism of a ring R, n be a positive integer and T_n（R, α） be the skew triangular matrix ring. We show that some properties related to nilpotent elements of R are inherited by T_n（R, α）. Meanwhile, we determine the strongly prime radical, generalized prime radical and Behrens radical of the ring R[x; α]/（x^n）, where R[x; α] is the skew polynomial ring.

On weakly nil-clean rings

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.

Exchange Rings Generated by Their Units

Let R be an exchange ring with primitive factors artinian. We prove that there exists a u∈U（R） such that 1R ± u ∈ U（R）, if and only if for any a ∈ R, there exists a u ∈ U（R） such that a ± u∈ U（R）. Phrthermore, we prove that, for any A ∈ Mn（R）（n ≥ 2）, there exists a U ∈ GLn（R） such that A ± U ∈ GLn（R）.