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Goldie extending elements in modular lattices. (English) Zbl 1424.06028
Summary: 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.

##### MSC:
 06C05 Modular lattices, Desarguesian lattices
##### Keywords:
modular lattice; Goldie extending element
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