On various prime and semiprime bi-ideals of rings

G. Mohanraj; M. Palanikumar

G. Mohanraj Annamalai University, Annamalainagar-608 002, Tamil Nadu, India.
M. Palanikumar Annamalai University, Annamalainagar-608 002, Tamil Nadu, India.

Abstract

In McCoy's fundamental paper \cite{Mccoy1} has the following theorem. \lq\lq If $P$ is an ideal in ring $R$, equivalent for the following conditions.\\
{P1:} $aRb \subseteq P \implies a \in P$ or $b \in P$.\\
{P2:} $I_1 I_2\subseteq P \implies I_1 \subseteq P$ or $I_2 \subseteq P$ for the right ideals $I_1$ and $I_2$ in $R$.\\
{P3:} $J_1 J_2 \subseteq P \implies J_1 \subseteq P$ or $J_2 \subseteq P$ for the left ideals $J_1$ and $J_2$ in $R$\rq\rq.\\
Of course for the prime right ideal, conditions {P1} and {P2} are equivalent and in the case of prime left ideal, conditions {P1} and {P3} are equivalent. However, we establish the prime bi-ideals conditions {P1}, {P2} and {P3} are entirely different. In this paper, three different prime bi-ideals are introduced that differentiate three conditions. A.P.J. van der Walt defined prime bi-ideal if P1 holds \cite {Van}.