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On Goldie absolute direct summands in modular lattices. (English) Zbl 07729576

Summary: Absolute direct summand in lattices is defined and some of its properties in modular lattices are studied. It is shown that in a certain class of modular lattices, the direct sum of two elements has absolute direct summand if and only if the elements are relatively injective. As a generalization of absolute direct summand (ADS for short), the concept of Goldie absolute direct summand in lattices is introduced and studied. It is shown that Goldie ADS property is inherited by direct summands. A necessary and sufficient condition is given for an element of modular lattice to have Goldie ADS.

MSC:

06B05 Structure theory of lattices
06C05 Modular lattices, Desarguesian lattices
06B99 Lattices
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