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.
Smith, P. F. (2014). COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES. International Electronic Journal of Algebra, 16(16), 16-31. https://doi.org/10.24330/ieja.266224
AMA
Smith PF. COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES. IEJA. December 2014;16(16):16-31. doi:10.24330/ieja.266224
Chicago
Smith, Patrick F. “COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES”. International Electronic Journal of Algebra 16, no. 16 (December 2014): 16-31. https://doi.org/10.24330/ieja.266224.
EndNote
Smith PF (December 1, 2014) COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES. International Electronic Journal of Algebra 16 16 16–31.
IEEE
P. F. Smith, “COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES”, IEJA, vol. 16, no. 16, pp. 16–31, 2014, doi: 10.24330/ieja.266224.
ISNAD
Smith, Patrick F. “COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES”. International Electronic Journal of Algebra 16/16 (December 2014), 16-31. https://doi.org/10.24330/ieja.266224.
JAMA
Smith PF. COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES. IEJA. 2014;16:16–31.
MLA
Smith, Patrick F. “COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES”. International Electronic Journal of Algebra, vol. 16, no. 16, 2014, pp. 16-31, doi:10.24330/ieja.266224.
Vancouver
Smith PF. COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES. IEJA. 2014;16(16):16-31.
