On Finitely Generated Module Whose First Nonzero Fitting Ideal is Prime

Abstract

Let R be a commutative ring and M be a finitely generated R-module. Let I(M) be the first nonzero Fitting ideal of M. The main result of this paper is to characterize modules whose first nonzero Fitting ideals are prime ideals, in some cases. As a consequence, it is shown that if M is an Artinian R-module and $${{\,\mathrm{I}\,}}(M) = \mathfrak {q}$$ is a prime ideal of R which contains a nonzero divisor, then $$M\cong R/\mathfrak {q}\oplus P,$$ for some submodule P of M.