ON WEAK MCCOY MODULES OVER COMMUTATIVE RINGS

Abstract

Let M be a module over a commutative ring R.  In this paper we generalize some annihilator conditions from rings to modules.  Denote by Nil(M) the set of all nilpotent elements of M.  M is said to be weak McCoy if f(x)m(x) = 0, where f(x) = \(\sum_{i=0}^{k}\) m_ixⁱ ∈ R[x]\{0} and m(x) = \(\sum_{j=0}^{n}\) a_jx^j ∈ M[x]\{0}, then sm_i ∈ Nil(M), for every 1 ≤ i ≤ k and for some s ∈ R\{0}.  We prove that the class of weak McCoy modules is closed under direct limit, finite direct product and localization.  We show that if R is a uniform R-module, then the direct sum of weak McCoy R-modules is again weak McCoy.  We prove that over a domain D, the D-module M is weak McCoy provided that if T(M) is weak McCoy, where T(M) is the torsion submodule of M.