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 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.