Mathematica Bohemica, Vol. 142, No. 2, pp. 163-180, 2017
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
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Affiliations: Shriram Khanderao Nimbhorkar, Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, Maharashtra, India, e-mail: firstname.lastname@example.org; Rupal Chandulal Shroff, Department of Core Engineering and Engineering Sciences, MIT College of Engineering, MIT College Road, Pune 411038, Maharashtra, India, e-mail: email@example.com