© Hindawi Publishing Corp. TWO VERSIONS OF NAKAYAMA LEMMA FOR MULTIPLICATION MODULES

The aim of this note is to generalize the Nakayama lemma to a class of multiplication mod-ules over commutative rings with identity. In this note, by considering the notion of multi-plication modules and the product of submodules of them, we state and prove two versions of Nakayama lemma for such modules. In the first version we give some equivalent condi-tions for faithful finitely generated multiplication modules, and in the second version we give them for faithful multiplication modules with a minimal generating set. 2000 Mathematics Subject Classification: 16D10. 1. Introduction. Let R be a commutative ring with identity and let M be a unitary R-module. Then M is called a multiplication R-module provided for each submod-ule N of M there exists an ideal I of R such that N = IM, we call I a presentation ideal of N. Note that our definitions agree with [2, 3], but in [5], the term multipli-cation module is used in a different way. (In this note, an R-module M is multipli