Abstract:
The purpose of this paper is to introduce some
new classes of modules which is closely related to the classes of
sharp modules, pseudo-Dedekind modules and T V -modules. In this
paper we introduce the concepts of Φ-sharp modules, Φ-pseudo-
Dedekind modules and Φ-T V -modules. Let R be a commutative
ring with identity and set H = {M | M is an R-module and Nil(M)
is a divided prime submodule of M}. For an R-module M ∈ H,
set T = (R \ Z(M)) ∩ (R \ Z(R)), T(M) = T
−1
(M) and P :=
(Nil(M) :R M). In this case the mapping Φ : T(M) −→ MP given
by Φ(x/s) = x/s is an R-module homomorphism. The restriction
of Φ to M is also an R-module homomorphism from M in to MP
given by Φ(m/1) = m/1 for every m ∈ M. An R-module M ∈ H
is called a Φ-sharp module if for every nonnil submodules N, L of
M and every nonnil ideal I of R with N ⊇ IL, there exist a nonnil
ideal I
′ ⊇ I of R and a submodule L
′ ⊇ L of M such that N = I
′L
′
.
We prove that Many of the properties and characterizations of sharp
modules may be extended to Φ-sharp modules, but some can not.
