Novelty for Different Prime Partial Bi-Ideals in Non-Commutative Partial Rings and Its Extension

M. Palanikumar, Omaima Al-Shanqiti, Chiranjibe Jana () and Madhumangal Pal
Additional contact information
M. Palanikumar: Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, India
Omaima Al-Shanqiti: Department of Applied Science, Umm Al-Qura University, Mecca P.O. Box 24341, Saudi Arabia
Chiranjibe Jana: Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India
Madhumangal Pal: Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India

Mathematics, 2023, vol. 11, issue 6, 1-11

Abstract: 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 m p b 1 , ( m p b 2 , m p b 3 ) systems is presented. In general, the m p b 2 system is a generalization of the m p b 1 system, while the m p b 3 system is a generalization of both m p b 2 and m p b 1 systems. If Φ is a prime bi-ideal of ℧, then Φ is a one-prime partial bi-ideal (two-prime partial bi-ideal, three-prime partial bi-ideal) if and only if ℧ \ Φ is a m p b 1 system ( m p b 2 system, m p b 3 system) of ℧. If Θ is a prime bi-ideal in the complete partial ring ℧ and Δ is an m p b 3 system of ℧ with Θ ∩ Δ = ϕ , then there exists a three-prime partial bi-ideal Φ of ℧, such that Θ ⊆ Φ with Φ ∩ Δ = ϕ . These are necessary and sufficient conditions for partial bi-ideal Θ to be a three-prime partial bi-ideal of ℧. It is shown that partial bi-ideal Θ is a three-prime partial bi-ideal of ℧ if and only if H Θ is a prime partial ideal of ℧. If Θ is a one-prime partial bi-ideal (two-prime partial bi-ideal) in ℧, then H Θ is a prime partial ideal of ℧. It is guaranteed that a three-prime partial bi-ideal Φ with a prime bi-ideal Θ does not meet the m p b 3 system. In order to strengthen our results, examples are provided.

Keywords: partial ring; prime bi-ideal; one-prime partial bi-ideal; two-prime partial bi-ideal; three-prime partial bi-ideal; m pb 1 system; m pb 2 system; m pb 3 system (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/11/6/1309/pdf (application/pdf)
https://www.mdpi.com/2227-7390/11/6/1309/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:11:y:2023:i:6:p:1309-:d:1091544

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().