Regularity of n-P-V-Rings and n-P-V’-Rings
The regularity of the n-P-V-rings and n-P-V’-rings is systematically investigated in this paper. Employing the notions of quasi-ideals, weakly left (or right) ideals, and generalized weak ideals, we focus on investigating the strong <inline-formula><math xmlns="http://www.w3.org/1998/M...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/13/12/863 |
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| Summary: | The regularity of the n-P-V-rings and n-P-V’-rings is systematically investigated in this paper. Employing the notions of quasi-ideals, weakly left (or right) ideals, and generalized weak ideals, we focus on investigating the strong <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regularity and weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regularity of the n-P-V-rings and the n-P-V’-rings. Subsequently, we demonstrate our results as follows: (1) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is strongly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regular if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is a left n-P-V-ring where all its maximal left ideals are either quasi-ideals, weakly right ideals, or generalized weak ideals. (2) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is strongly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regular iff <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is an abelian left (right) n-P-V’-ring where all its maximal essential left (right) ideals are either quasi-ideals, weakly right (left) ideals, or generalized weak ideals. (3) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is reduced left weakly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi></mrow></semantics></math></inline-formula>-regular if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi></mrow></semantics></math></inline-formula> is an idempotent reflexive semi-abelian left n-P-V’-ring where all its maximal essential left ideals are either quasi-ideals, weakly right ideals, or generalized weak ideals. |
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| ISSN: | 2075-1680 |