The exchange rings without unity, first introduced by Ara, are further investigated. Some new characterizations and properties of exchange general rings are given. For example, a general ring I is exchange if and only...The exchange rings without unity, first introduced by Ara, are further investigated. Some new characterizations and properties of exchange general rings are given. For example, a general ring I is exchange if and only if for any left ideal L of I and a^-= a^-2 ∈I/L, there exists w ∈ r. ureg（I） such that w^- = a^-; E（R, I） （ the ideal extension of a ring R by its ideal I） is an exchange ring if and only if R and I are both exchange. Furthermore, it is presented that if I is a two-sided ideal of a unital ring R and I is an exchange general ring, then every central element of I is a clean element in 1.展开更多
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]...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.展开更多
基金The National Natural Science Foundation of China(No10571026),the Natural Science Foundation of Jiangsu Province(NoBK2005207), the Teaching and Research Award Program for Out-standing Young Teachers in Higher Education Institutions of MOE,PRC
文摘The exchange rings without unity, first introduced by Ara, are further investigated. Some new characterizations and properties of exchange general rings are given. For example, a general ring I is exchange if and only if for any left ideal L of I and a^-= a^-2 ∈I/L, there exists w ∈ r. ureg（I） such that w^- = a^-; E（R, I） （ the ideal extension of a ring R by its ideal I） is an exchange ring if and only if R and I are both exchange. Furthermore, it is presented that if I is a two-sided ideal of a unital ring R and I is an exchange general ring, then every central element of I is a clean element in 1.
基金The National Natural Science Foundation （10571026） of China, and the Natural Science Foundation （BK2005207） of Jiangsu Province.
文摘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.
