Zip and weak zip algebras in a congruence-modular variety
The zip (commutative) rings, introduced by Faith and Zelmanowitz, generated a fruitful line of investigation in ring theory. Recently, Dube, Blose and Taherifar developed an abstract theory of zippedness by means of frames. Starting from some ideas contained in their papers, we define and study the...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Shahid Bahonar University of Kerman
2024-12-01
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| Series: | Journal of Mahani Mathematical Research |
| Subjects: | |
| Online Access: | https://jmmrc.uk.ac.ir/article_4410_1192ef9e4e67f8ad0230c8ba6c5cd119.pdf |
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| Summary: | The zip (commutative) rings, introduced by Faith and Zelmanowitz, generated a fruitful line of investigation in ring theory. Recently, Dube, Blose and Taherifar developed an abstract theory of zippedness by means of frames. Starting from some ideas contained in their papers, we define and study the zip and weak zip algebras in a semidegenerate congruence-modular variety $\mathcal{V}$. We obtain generalizations of some results existing in the literature of zip rings and zipped frames. For example, we prove that a neo-commutative algebra $A\in \mathcal{V}$ is a weak zip algebra if and only if the frame $RCon(A)$ of radical congruences of $A$ is a zipped frame (in the sense of Dube and Blose). We study the way in which the reticulation functor preserves the zippedness property. Using the reticulation and a Hochster's theorem we prove that a neo-commutative algebra $A\in \mathcal{V}$ is a weak zip algebra if and only if the minimal prime spectrum $Min(A)$ of $A$ is a finite space. |
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| ISSN: | 2251-7952 2645-4505 |