拟Abel环

设R是一个环,M是双R-模.若对每个e∈E(R),有eR(1-e)Me=eM(1-e)Re=0,则称M为拟Abel模,这里E(R)表示R的幂等元集合.若R-双模R是拟Abel的,则称R为拟Abel环.证明了如下结果:①R为拟Abel环当且仅当对任意的a∈N(R),e∈E(R),ea=0蕴涵eRae=0,这里N(R)表示R的幂零元集合;②R为Abel环当且仅当R为幂零自反环和拟Abel环;③设σ为环R的环自同态映射且满足条件: e∈E(R),σ(e)=e,则R为拟Abel环当且仅当R(σ)为拟Abel模.

Quasi-Abel rings

Let R be a ring and M a R-bimodule. If eR(1- e)Me = eM(1- e)Re = 0 for every element e∈E E(R), then M is called quasi-Abel module, where E(R) is the set of all idempotent elements of R. If R as R-bimodule is quasi-Abel module, then R is called a quasi-Abel ring. This article proves some main results as follows: ① R is a quasi-Abel ring if and only if for any element a∈N(R), e∈ E(R), ea = 0 implies eRae=0; ② R is an Abel ring if and only if R is a nilpotent reflexive ring and quasi-Abel ring; (3) Suppose σ is an endomorphism of ring R such that σ(e) =e for any e ∈ E(R) , then R is a quasi-Abel ring if and only if R(σ) is quasi-Abel module.