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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
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