Submodules of multiplication modules (Q812864)

Let \(R\) be a commutative ring with identity and \(M\) an (unitary) \(R\)-module. We say that \(M\) is multiplication module if for each submodule \(N\) of \(M,N = IM\) for some ideal \(I\) of \(R\). In this paper, the author studies the set \(S(M)\) of finitely generated faithful multiplication submodules of a finitely generated faithful multiplication module \(M\) and generalizes results about finitely generated faithful multiplication ideals due to \textit{M. M. Ali} and \textit{D. J. Smith} [Beitr. Algebra Geom. 42, No. 1, 219--233 (2001; Zbl 0971.13002)] to finitely generated faithful multiplication modules. In this context, \(S(M)\) is a multiplicative semigroup with multiplication defined for \(K, N\in S(M)\) by \(KN= IJM\), where \(K= IM\) and \(N= JM\) for ideals \(I\) and \(J\) of \(R\), and divisibility in \(S(M)\) is defined by \(K|N\) if \(K\subseteq N\). The author then investigates the relationship between greatest common divisors and least common multiples in \(S(M)\).