Modules: Duals and principally generated fake duals
type: Item
instance of: scholarly article
title: Modules: Duals and principally generated fake duals
zbMATH Open document ID: 0651.06006
DOI: 10.1007/BF01188388
author: E. W. Johnson
published in: Algebra Universalis
publication date: 1987-01-01T00:00:00Z
review text: Let M be a module over the commutative Noetherian ring R (with identity). Observations of \textit{D. D. Anderson} [J. Algebra 47, 425-432 (1977; Zbl 0383.13011)] concerning the behaviour of the lattice dual of the submodule lattice of M lead the author, in the paper under review, to study a Noether lattice \({\mathcal L}\) in the sense of \textit{R. P. Dilworth} [Pac. J. Math. 12, 481-498 (1962; Zbl 0111.041)] and a complete modular lattice L on which there is defined a multiplication by elements of \({\mathcal L}\) satisfying certain properties. On the assumption that L is principally generated and that \({\mathcal L}\) is L-principally generated, the author investigates conditions under which the lattice dual \(L^*\) is principally generated. There are consequences in commutative algebra, including an algebraic characterization of complete local rings.
Mathematics Subject Classification ID: 06F10
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 13J10
zbMATH DE Number: 4061305
zbMATH Keywords: algebraic lattice
zbMATH Keywords: commutative Noetherian ring
zbMATH Keywords: complete semilocal ring
zbMATH Keywords: Noetherian module
zbMATH Keywords: Artinian module
zbMATH Keywords: principally generated lattice
zbMATH Keywords: lattice dual of the submodule lattice
zbMATH Keywords: Noether lattice
zbMATH Keywords: complete modular lattice
zbMATH Keywords: complete local rings
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4061305