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Modules Whose Cyclic Submodules Have Finite Dimension

Published online by Cambridge University Press:  20 November 2018

David Berry
David Berry*
Affiliation:
University of Kentucky, Lexington, Kentucky 40506, U.S.A.
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R denotes an associative ring with identity. Module means unitary right R-module. A module has finite Goldie dimension over R if it does not contain an infinite direct sum of nonzero submodules [6]. We say R has finite (right) dimension if it has finite dimension as a right R-module. We denote the fact that M has finite dimension by dim (M)<∞.

A nonzero submodule N of a module M is large in M if N has nontrivial intersection with nonzero submodules of M [7]. In this case M is called an essential extension of N. N⊆′M will denote N is essential (large) in M. If N has no proper essential extension in M, then N is closed in M. An injective essential extension of M, denoted I(M), is called the injective hull of M.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 1 , 01 March 1976 , pp. 1 - 6
Copyright
Copyright © Canadian Mathematical Society 1976

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