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## COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES

#### Patrick F. Smith [1]

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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.
Lattice of ideals, lattice of submodules, multiplication modules, complete lattice, complete homomorphism
Konular JA38YU69RD Makaleler Yazar: Patrick F. Smith
 Bibtex @ { ieja266224, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {Prof. Dr. Abdullah HARMANCI}, year = {2014}, volume = {16}, pages = {16 - 31}, doi = {10.24330/ieja.266224}, title = {COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES}, key = {cite}, author = {Smith, Patrick F.} } APA Smith, P . (2014). COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES. International Electronic Journal of Algebra, 16 (16), 16-31. DOI: 10.24330/ieja.266224 MLA Smith, P . "COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES". International Electronic Journal of Algebra 16 (2014): 16-31 Chicago Smith, P . "COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES". International Electronic Journal of Algebra 16 (2014): 16-31 RIS TY - JOUR T1 - COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES AU - Patrick F. Smith Y1 - 2014 PY - 2014 N1 - doi: 10.24330/ieja.266224 DO - 10.24330/ieja.266224 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 16 EP - 31 VL - 16 IS - 16 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.266224 UR - http://dx.doi.org/10.24330/ieja.266224 Y2 - 2019 ER - EndNote %0 International Electronic Journal of Algebra COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES %A Patrick F. Smith %T COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES %D 2014 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 16 %N 16 %R doi: 10.24330/ieja.266224 %U 10.24330/ieja.266224 ISNAD Smith, Patrick F. . "COMPLETE HOMOMORPHISMS BETWEEN MODULE LATTICES". International Electronic Journal of Algebra 16 / 16 (Aralık 2014): 16-31. http://dx.doi.org/10.24330/ieja.266224