Strongly Fully Stable Acts Relative to an Ideal

Abstract

The purpose of this paper is to introduce and investigate strongly fully stable acts relative to an ideal as a concept generalizing strongly fully stable modules relative to an ideal, but is stronger than that of fully stable acts. In this study, we consider some properties and characterizations of the class of strongly fully stable acts relative to an ideal, as well as the relations between this class and other classes. Among these classes are quasi-injective acts, strongly quasi-injective acts, acts which satisfy Baer's criterion, acts satisfying the strongly Baer's criterion, and duo acts. The product of strongly fully stable acts relative to an ideal need not be a strongly fully stable act relative to that ideal. The coproduct of any family of strongly fully stable acts relative to an ideal need not be a strongly fully stable act relative to that ideal. Also, we have that strongly fully stable acts relative to an ideal are equivalent to an $S$-act that satisfies the strongly Baer's criterion relative to an ideal $I$ for cyclic subacts. The strongly fully stable act relative to $I$ is equivalent to the strongly quasi-injective act relative to the ideal $I$ and duo act with a commutative monoid.