A NOTE ON $\sigma(*)$-RINGS AND THEIR EXTENSIONS
Let <i>R </i>be an associative ring with identity 1 <i>6</i>= 0, and <i> </i>an endomorphism of <i>R</i>. We recall (<i>¤</i>) property on <i>R </i>(i.e. <i>a.</i>(<i>a</i>) <i>2 P</i>(<i>R</i>) implies <i>a 2 P</i>(<i>R</i>) for <i>a 2 R</i>, where <i>P</i>(<i>R</i>) is the prime radical of <i>R</i>). Also recall that a ring <i>R </i>is said to be 2-primal if and only if the prime radical <i>P</i>(<i>R</i>) and nil radical are same, i.e. if the prime radical is a completely semiprime ideal. It can be seen that a <i>.</i>(<i>¤</i>) ring is a 2-primal ring. Let <i>R </i>be a ring and <i>. </i>an automorphism of <i>R</i>. Then we know that <i>. </i>can be extended to an automorphism of the skew polynomial ring <i>R</i>[<i>x</i>; <i>.</i>]. In this paper we show that if <i>R </i>is a Noetherian ring and <i>. </i>is an automorphism of <i>R </i>such that <i>R </i>is a <i>.</i>(<i>¤</i>)-ring, then <i>R</i>[<i>x</i>; <i>.</i>] is also a <i>.</i>(<i>¤</i>)-ring.