Mathematica Bohemica, online first, 18 pp.

Goldie extending elements in modular lattices

Shriram K. Nimbhorkar, Rupal C. Shroff

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

Shriram Khanderao Nimbhorkar, Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, Maharashtra, India, e-mail:; Rupal Chandulal Shroff, Department of Core Engineering and Engineering Sciences, MIT College of Engineering, MIT College Road, Pune 411038, Maharashtra, India, e-mail:

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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