The purpose of this paper is to generalize ([1], Theorem 1.22) and ([2] , Proposition 1(1)). We obtain;Let R be a ring, and let Q=EndR(M), where M is a generalized quasi-injective left R-module. Then( 1 ) J(Q) =Z(Q);(...The purpose of this paper is to generalize ([1], Theorem 1.22) and ([2] , Proposition 1(1)). We obtain;Let R be a ring, and let Q=EndR(M), where M is a generalized quasi-injective left R-module. Then( 1 ) J(Q) =Z(Q);( 2 ) Q/J(Q) is a Von Neumann regular ring.展开更多
Since several years, Artinian semisirnple rings have drawn the attention of various authors (cf. for example, [l]-[10]). The purpose of this paper is to characterize Artinian semisimple rings in terms of P- in jectivi...Since several years, Artinian semisirnple rings have drawn the attention of various authors (cf. for example, [l]-[10]). The purpose of this paper is to characterize Artinian semisimple rings in terms of P- in jectivity. New characteri-zations of Artinian semisimple rings are obtained. Necessary and sufficient con-ditions for prime rings to be Artinian simple are given. Several interesting pro-perties of P- injective rings are derived.展开更多
In [1] the following problem was proposed: Is a ring R strongly regular if every ideal of R is idempotent and every maximal left ideal of R is an ideal? In [2], it was asked whether a ring R is von Neumann regular if ...In [1] the following problem was proposed: Is a ring R strongly regular if every ideal of R is idempotent and every maximal left ideal of R is an ideal? In [2], it was asked whether a ring R is von Neumann regular if every ideal of R is idempotent and every maximal essential left ideal of R is an ideal? The primary aim of this note is to construct a counterexample for the above questions.展开更多
文摘The purpose of this paper is to generalize ([1], Theorem 1.22) and ([2] , Proposition 1(1)). We obtain;Let R be a ring, and let Q=EndR(M), where M is a generalized quasi-injective left R-module. Then( 1 ) J(Q) =Z(Q);( 2 ) Q/J(Q) is a Von Neumann regular ring.
文摘Since several years, Artinian semisirnple rings have drawn the attention of various authors (cf. for example, [l]-[10]). The purpose of this paper is to characterize Artinian semisimple rings in terms of P- in jectivity. New characteri-zations of Artinian semisimple rings are obtained. Necessary and sufficient con-ditions for prime rings to be Artinian simple are given. Several interesting pro-perties of P- injective rings are derived.
基金Project supported by the Grant of Anhui Education Council
文摘In [1] the following problem was proposed: Is a ring R strongly regular if every ideal of R is idempotent and every maximal left ideal of R is an ideal? In [2], it was asked whether a ring R is von Neumann regular if every ideal of R is idempotent and every maximal essential left ideal of R is an ideal? The primary aim of this note is to construct a counterexample for the above questions.
