ML-环

称环R为左ML-环,若环R中任意元a满足a或1-a是左Morphic元.显然,左Morphic环及局部环皆为左ML-环,但反之不然.设{Ri}i∈I是环族.得到的∏i∈IRi是左ML-环当且仅当存在i0∈I使得Ri0是左ML-环且对任意i∈I-{i0},Ri都是左Morphic环.此外,若正整数n≥2且n=∏si=1prii是n的标准因子分解,则Zn∝Zn是左ML-环当且仅当至多一个i使得ri>1当且仅当Zn是VNL-环.同时还构造了一些例子来说明问题.

On ML-Rings

A ring R is called a left ML-ring if a or 1-a is left morphic for every a∈R.Left morphic rings and local rings are left ML-rings but conversely is not true.Let R i(i∈I) be rings.It is shown that ∏ i∈I R i is a left ML-ring if and only if there exists i 0 ∈I such that R i 0 is a left ML-ring and for each i∈I-{ i 0 },R i is a left morphic ring.Moreover,if n≥2 and n = ∏ s i = 1 p r i i is a prime power decomposition of n,then Z n ∝Z n is a left ML-ring if and only if r i > 1 for at most one value of i if and only if Z n is a VNL-ring.Some examples were also given.