In this paper some results on direct summands of Goldie extending elements are studied in a modular lattice. An element of a lattice with is said to be a Goldie extending element if and only if for every there exists a direct summand of such that is essential in both and . Some characterizations of decomposition of a Goldie extending element in a modular lattice are obtained.
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...
The concept of an extending ideal in a modular lattice is introduced. A translation of module-theoretical concept of ojectivity (i.e. generalized relative injectivity) in the context of the lattice of ideals of a modular lattice is introduced. In a modular lattice satisfying a certain condition, a characterization is given for direct summands of an extending ideal to be mutually ojective. We define exchangeable decomposition and internal exchange property of an ideal in a modular lattice. It is...
The concept of a Goldie extending module is generalized to a Goldie extending element in a lattice. An element of a lattice with is said to be a Goldie extending element if and only if for every there exists a direct summand of such that is essential in both and . 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...
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