Goldie extending elements in modular lattices
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 bo...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2017-07-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/142/2/mb142_2_5.pdf |
| 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. |
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| ISSN: | 0862-7959 2464-7136 |