On Study of Monoids in Which Right Ideals are Generators

Abstract

The main purpose of this paper is to study some classes of monoids in which every right ideal is a generator. We describe commutative monoids with this property and specially we show that for a commutative monoid S, every ideal is a generator if and only if S is a cancellative hereditary monoid which is equivalent to $$S=G$$ or $$S=G\times A$$ where G is a group and A is the additive monoid of non-negative integers. Also it is shown that for a non-trivial commutative monoid S without zero any ideal is a generator if and only if there exists a free S-act A such that for any subact B of $$A, trace~(B,A)=A$$ . Moreover for a regular monoid S it is shown any right ideal of S is a generator if and only if S has no two sided ideal.