Abstract

Let N be a right near-ring. N is said to be strongly reduced if, for <TEX>$a\inN$</TEX>, <TEX>$a^2 \in N_{c}$</TEX> implies <TEX>$a\;\in\;N_{c}$</TEX>, or equivalently, for <TEX>$a\inN$</TEX> and any positive integer n, <TEX>$a^{n} \in N_{c}$</TEX> implies <TEX>$a\;\in\;N_{c}$</TEX>, where <TEX>$N_{c}$</TEX> denotes the constant part of N. We will show that strong reducedness is equivalent to condition (ⅱ) of Reddy and Murty's property <TEX>$(^{\ast})$</TEX> (cf. [Reddy & Murty: On strongly regular near-rings. Proc. Edinburgh Math. Soc. (2) 27 (1984), no. 1, 61-64]), and that condition (ⅰ) of Reddy and Murty's property <TEX>$(^{\ast})$</TEX> follows from strong reducedness. Also, we will investigate some characterizations of strongly reduced near-rings and their properties. Using strong reducedness, we characterize left strongly regular near-rings and (<TEX>$P_{0}$</TEX>)-near-rings.