ABSTRACT

We introduce the concept of weak module systems for commutative monoids. This concept is a common generalization of the notion of a weak ideal system as presented in the author’s book “Ideal Systems” (M. Dekker, 1998) and the notion of a module system as presented in the author’s article “Localizing Systems, Module Systems, and Semistar Operations” (J. Algebra 238, 2001). With the aid of this concept, we first develop a purely multiplicative theory of integral elements, valid for commutative monoids and rings without any cancellation assumptions. Next, we generalize the Marot property (used in the theory of rings with zero divisors) to monoids and shed new light on the theory of Dedekind and Prüfer monoids without cancellation.