Novelty for Different Prime Partial Bi-Ideals in Non-Commutative Partial Rings and Its Extension
In computer programming languages, partial additive semantics are used. Since partial functions under disjoint-domain sums and functional composition do not constitute a field, linear algebra cannot be applied. A partial ring can be viewed as an algebraic structure that can process natural partial o...
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MDPI AG
2023-03-01
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| author | M. Palanikumar Omaima Al-Shanqiti Chiranjibe Jana Madhumangal Pal |
| author_facet | M. Palanikumar Omaima Al-Shanqiti Chiranjibe Jana Madhumangal Pal |
| author_sort | M. Palanikumar |
| collection | DOAJ |
| description | In computer programming languages, partial additive semantics are used. Since partial functions under disjoint-domain sums and functional composition do not constitute a field, linear algebra cannot be applied. A partial ring can be viewed as an algebraic structure that can process natural partial orderings, infinite partial additions, and binary multiplications. In this paper, we introduce the notions of a one-prime partial bi-ideal, a two-prime partial bi-ideal, and a three-prime partial bi-ideal, as well as their extensions to partial rings, in addition to some characteristics of various prime partial bi-ideals. In this paper, we demonstrate that two-prime partial bi-ideal is a generalization of a one-prime partial bi-ideal, and three-prime partial bi-ideal is a generalization of a two-prime partial bi-ideal and a one-prime partial bi-ideal. A discussion of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>1</mn></mrow></msub><mo>,</mo><mrow><mo>(</mo><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>3</mn></mrow></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> systems is presented. In general, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>2</mn></mrow></msub></semantics></math></inline-formula> system is a generalization of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>1</mn></mrow></msub></semantics></math></inline-formula> system, while the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>3</mn></mrow></msub></semantics></math></inline-formula> system is a generalization of both <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>2</mn></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>1</mn></mrow></msub></semantics></math></inline-formula> systems. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Φ</mi></semantics></math></inline-formula> is a prime bi-ideal of ℧, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Φ</mi></semantics></math></inline-formula> is a one-prime partial bi-ideal (two-prime partial bi-ideal, three-prime partial bi-ideal) if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>℧</mo><mo>\</mo><mi mathvariant="normal">Φ</mi></mrow></semantics></math></inline-formula> is a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>1</mn></mrow></msub></semantics></math></inline-formula> system (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>2</mn></mrow></msub></semantics></math></inline-formula> system, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>3</mn></mrow></msub></semantics></math></inline-formula> system) of ℧. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Θ</mi></semantics></math></inline-formula> is a prime bi-ideal in the complete partial ring ℧ and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Δ</mi></semantics></math></inline-formula> is an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>3</mn></mrow></msub></semantics></math></inline-formula> system of ℧ with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo>∩</mo><mi mathvariant="normal">Δ</mi><mo>=</mo><mi>ϕ</mi></mrow></semantics></math></inline-formula>, then there exists a three-prime partial bi-ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Φ</mi></semantics></math></inline-formula> of ℧, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo>⊆</mo><mi mathvariant="normal">Φ</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">Φ</mi><mo>∩</mo><mi mathvariant="normal">Δ</mi><mo>=</mo><mi>ϕ</mi></mrow></semantics></math></inline-formula>. These are necessary and sufficient conditions for partial bi-ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Θ</mi></semantics></math></inline-formula> to be a three-prime partial bi-ideal of ℧. It is shown that partial bi-ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Θ</mi></semantics></math></inline-formula> is a three-prime partial bi-ideal of ℧ if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi mathvariant="normal">Θ</mi></msub></semantics></math></inline-formula> is a prime partial ideal of ℧. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Θ</mi></semantics></math></inline-formula> is a one-prime partial bi-ideal (two-prime partial bi-ideal) in ℧, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi mathvariant="normal">Θ</mi></msub></semantics></math></inline-formula> is a prime partial ideal of ℧. It is guaranteed that a three-prime partial bi-ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Φ</mi></semantics></math></inline-formula> with a prime bi-ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Θ</mi></semantics></math></inline-formula> does not meet the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>3</mn></mrow></msub></semantics></math></inline-formula> system. In order to strengthen our results, examples are provided. |
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| spelling | doaj.art-c65e899e2de844e0b82c137831f42ebd2023-11-17T12:26:48ZengMDPI AGMathematics2227-73902023-03-01116130910.3390/math11061309Novelty for Different Prime Partial Bi-Ideals in Non-Commutative Partial Rings and Its ExtensionM. Palanikumar0Omaima Al-Shanqiti1Chiranjibe Jana2Madhumangal Pal3Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, IndiaDepartment of Applied Science, Umm Al-Qura University, Mecca P.O. Box 24341, Saudi ArabiaDepartment of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, IndiaDepartment of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, IndiaIn computer programming languages, partial additive semantics are used. Since partial functions under disjoint-domain sums and functional composition do not constitute a field, linear algebra cannot be applied. A partial ring can be viewed as an algebraic structure that can process natural partial orderings, infinite partial additions, and binary multiplications. In this paper, we introduce the notions of a one-prime partial bi-ideal, a two-prime partial bi-ideal, and a three-prime partial bi-ideal, as well as their extensions to partial rings, in addition to some characteristics of various prime partial bi-ideals. In this paper, we demonstrate that two-prime partial bi-ideal is a generalization of a one-prime partial bi-ideal, and three-prime partial bi-ideal is a generalization of a two-prime partial bi-ideal and a one-prime partial bi-ideal. A discussion of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>1</mn></mrow></msub><mo>,</mo><mrow><mo>(</mo><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>3</mn></mrow></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> systems is presented. In general, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>2</mn></mrow></msub></semantics></math></inline-formula> system is a generalization of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>1</mn></mrow></msub></semantics></math></inline-formula> system, while the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>3</mn></mrow></msub></semantics></math></inline-formula> system is a generalization of both <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>2</mn></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>1</mn></mrow></msub></semantics></math></inline-formula> systems. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Φ</mi></semantics></math></inline-formula> is a prime bi-ideal of ℧, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Φ</mi></semantics></math></inline-formula> is a one-prime partial bi-ideal (two-prime partial bi-ideal, three-prime partial bi-ideal) if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>℧</mo><mo>\</mo><mi mathvariant="normal">Φ</mi></mrow></semantics></math></inline-formula> is a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>1</mn></mrow></msub></semantics></math></inline-formula> system (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>2</mn></mrow></msub></semantics></math></inline-formula> system, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>3</mn></mrow></msub></semantics></math></inline-formula> system) of ℧. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Θ</mi></semantics></math></inline-formula> is a prime bi-ideal in the complete partial ring ℧ and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Δ</mi></semantics></math></inline-formula> is an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>3</mn></mrow></msub></semantics></math></inline-formula> system of ℧ with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo>∩</mo><mi mathvariant="normal">Δ</mi><mo>=</mo><mi>ϕ</mi></mrow></semantics></math></inline-formula>, then there exists a three-prime partial bi-ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Φ</mi></semantics></math></inline-formula> of ℧, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo>⊆</mo><mi mathvariant="normal">Φ</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">Φ</mi><mo>∩</mo><mi mathvariant="normal">Δ</mi><mo>=</mo><mi>ϕ</mi></mrow></semantics></math></inline-formula>. These are necessary and sufficient conditions for partial bi-ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Θ</mi></semantics></math></inline-formula> to be a three-prime partial bi-ideal of ℧. It is shown that partial bi-ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Θ</mi></semantics></math></inline-formula> is a three-prime partial bi-ideal of ℧ if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi mathvariant="normal">Θ</mi></msub></semantics></math></inline-formula> is a prime partial ideal of ℧. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Θ</mi></semantics></math></inline-formula> is a one-prime partial bi-ideal (two-prime partial bi-ideal) in ℧, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi mathvariant="normal">Θ</mi></msub></semantics></math></inline-formula> is a prime partial ideal of ℧. It is guaranteed that a three-prime partial bi-ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Φ</mi></semantics></math></inline-formula> with a prime bi-ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Θ</mi></semantics></math></inline-formula> does not meet the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mrow><mi>p</mi><mi>b</mi><mn>3</mn></mrow></msub></semantics></math></inline-formula> system. In order to strengthen our results, examples are provided.https://www.mdpi.com/2227-7390/11/6/1309partial ringprime bi-idealone-prime partial bi-idealtwo-prime partial bi-idealthree-prime partial bi-ideal<i>m</i><sub><i>pb</i>1</sub> system |
| spellingShingle | M. Palanikumar Omaima Al-Shanqiti Chiranjibe Jana Madhumangal Pal Novelty for Different Prime Partial Bi-Ideals in Non-Commutative Partial Rings and Its Extension Mathematics partial ring prime bi-ideal one-prime partial bi-ideal two-prime partial bi-ideal three-prime partial bi-ideal <i>m</i><sub><i>pb</i>1</sub> system |
| title | Novelty for Different Prime Partial Bi-Ideals in Non-Commutative Partial Rings and Its Extension |
| title_full | Novelty for Different Prime Partial Bi-Ideals in Non-Commutative Partial Rings and Its Extension |
| title_fullStr | Novelty for Different Prime Partial Bi-Ideals in Non-Commutative Partial Rings and Its Extension |
| title_full_unstemmed | Novelty for Different Prime Partial Bi-Ideals in Non-Commutative Partial Rings and Its Extension |
| title_short | Novelty for Different Prime Partial Bi-Ideals in Non-Commutative Partial Rings and Its Extension |
| title_sort | novelty for different prime partial bi ideals in non commutative partial rings and its extension |
| topic | partial ring prime bi-ideal one-prime partial bi-ideal two-prime partial bi-ideal three-prime partial bi-ideal <i>m</i><sub><i>pb</i>1</sub> system |
| url | https://www.mdpi.com/2227-7390/11/6/1309 |
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