Abstract
We examine the properties of certain mappings between the lattice
L(R) of ideals of a commutative ring R and the lattice L(RM) of submodules
of an R-module M, in particular considering when these mappings are complete
homomorphisms of the lattices. We prove that the mapping λ from L(R)
to L(RM) defined by λ(B) = BM for every ideal B of R is a complete homomorphism
if M is a faithful multiplication module. A ring R is semiperfect
(respectively, a finite direct sum of chain rings) if and only if this mapping
λ : L(R) → L(RM) is a complete homomorphism for every simple (respectively,
cyclic) R-module M. A Noetherian ring R is an Artinian principal ideal
ring if and only if, for every R-module M, the mapping λ : L(R) → L(RM) is
a complete homomorphism.
Abstract
Details
| Other ID | JA38YU69RD |
|---|---|
| Journal Section | Articles |
| Authors | |
| Publication Date | December 1, 2014 |
| Published in Issue | Year 2014 Volume: 16 Issue: 16 |