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: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics of the Czech Academy of Science
2017-07-01
|
| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/142/2/mb142_2_5.pdf |
| _version_ | 1818877432898256896 |
|---|---|
| author | Shriram K. Nimbhorkar Rupal C. Shroff |
| author_facet | Shriram K. Nimbhorkar Rupal C. Shroff |
| author_sort | Shriram K. Nimbhorkar |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-19T13:58:12Z |
| format | Article |
| id | doaj.art-79142b3d12ed44909fd16b8b2ccccc4d |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-12-19T13:58:12Z |
| publishDate | 2017-07-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-79142b3d12ed44909fd16b8b2ccccc4d2022-12-21T20:18:32ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362017-07-01142216318010.21136/MB.2016.0049-14MB.2016.0049-14Goldie extending elements in modular latticesShriram K. NimbhorkarRupal C. ShroffThe 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.http://mb.math.cas.cz/full/142/2/mb142_2_5.pdf modular lattice Goldie extending element |
| spellingShingle | Shriram K. Nimbhorkar Rupal C. Shroff Goldie extending elements in modular lattices Mathematica Bohemica modular lattice Goldie extending element |
| title | Goldie extending elements in modular lattices |
| title_full | Goldie extending elements in modular lattices |
| title_fullStr | Goldie extending elements in modular lattices |
| title_full_unstemmed | Goldie extending elements in modular lattices |
| title_short | Goldie extending elements in modular lattices |
| title_sort | goldie extending elements in modular lattices |
| topic | modular lattice Goldie extending element |
| url | http://mb.math.cas.cz/full/142/2/mb142_2_5.pdf |
| work_keys_str_mv | AT shriramknimbhorkar goldieextendingelementsinmodularlattices AT rupalcshroff goldieextendingelementsinmodularlattices |