COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES

Authors : Patrick F Smith

Pages : 16-31

Doi:10.24330/ieja.266224

View : 25 | Download : 9

Publication Date : 2014-12-01

Article Type : Research Paper

Abstract :We examine the properties of certain mappings between the lattice Linsert ignore into journalissuearticles values(R); of ideals of a commutative ring R and the lattice Linsert ignore into journalissuearticles values(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 Linsert ignore into journalissuearticles values(R); to Linsert ignore into journalissuearticles values(RM); defined by λinsert ignore into journalissuearticles values(B); = BM for every ideal B of R is a complete homomorphism if M is a faithful multiplication module. A ring R is semiperfect insert ignore into journalissuearticles values(respectively, a finite direct sum of chain rings); if and only if this mapping λ : Linsert ignore into journalissuearticles values(R); → Linsert ignore into journalissuearticles values(RM); is a complete homomorphism for every simple insert ignore into journalissuearticles values(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 : Linsert ignore into journalissuearticles values(R); → Linsert ignore into journalissuearticles values(RM); is a complete homomorphism.
Keywords : Lattice of ideals, lattice of submodules, multiplication modules, complete lattice, complete homomorphism