Communications of the
Korean Mathematical Society CKMS

In this paper we introduce the notion of $P$-strongly regular near-ring. We have shown that a zero-symmetric near-ring $N$ is $P$-strongly regular if and only if $N$ is $P$-regular and $P$ is a completely semiprime ideal. We have also shown that in a $P$-strongly regular near-ring $N$, the following holds: (i) $Na+P$ is an ideal of $N$ for any $a \in N$. (ii) Every $P$-prime ideal of $N$ containing $P$ is maximal. (iii) Every ideal $I$ of $N$ fulfills $I+P=I^{2}+P$.

Keywords: $P$-regular, $P$-strongly regular, $P$-prime ideal, completely semiprime ideal

MSC numbers: 16Y30, 16Y60