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Goldie extending elements in modular lattices

Shriram K. Nimbhorkar, Rupal C. Shroff

Received June 28, 2014.   First published December 2, 2016.

Abstract: The concept of a Goldie extending module is generalized to a Goldie extending element in a lattice. An element $a$ of a lattice $L$ with $0$ is said to be a Goldie extending element if and only if for every $b \leq a$ there exists a direct summand $c$ of $a$ such that $b \wedge c$ is essential in both $b$ and $c$. Some properties of such elements are obtained in the context of modular lattices. We give a necessary condition for the direct sum of Goldie extending elements to be Goldie extending. Some characterizations of a decomposition of a Goldie extending element in such a lattice are given. The concepts of an $a$-injective and an $a$-ejective element are introduced in a lattice and their properties related to extending elements are discussed.

Keywords: modular lattice; Goldie extending element

Classification (MSC 2010): 06B10, 06C05

DOI: 10.21136/MB.2016.0049-14

Full text available as PDF.

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