P-Regular Nearrings Characterized by Their Bi-ideals

TL;DR

This paper extends the concept of bi-ideals in P-regular nearrings, providing new characterizations and representations of elements within these algebraic structures.

Contribution

It generalizes bi-ideals in P-regular nearrings and offers new characterizations and element representations within these structures.

Findings

Every element of a bi-ideal can be expressed as a sum of elements from P and Q.

Elements in the intersection of bi-ideals can be represented as sums involving P and products of bi-ideals.

The paper provides structural insights into P-regular nearrings and their bi-ideals.

Abstract

Using the idea of quasi-ideals of $P$-regular nearrings, the concept of bi-ideals of $P$-regular nearrings is generalized, which is an extension of the concept of quasi-ideals of $P$-regular nearrings and some interesting characterizations of bi-ideals are obtained. As a result, we prove that every element of a bi-ideal $B$ of a $P$-regular nearring can be represented as the sum of two elements of $P$ and $Q$. Moreover, every element of the finite intersection $\displaystyle \bigcap_{i=1}^{n}B_{i}$ of bi-ideals of a $P$-regular distributive nearring $N$ can be represented as the sum of two elements of $P$ and $B_{1}NB_{2}N...NB_{n-1}NB_{n}$.