A Construction in General Radical Theory

Abstract

Given an arbitrary associative ring R we consider the ring R [ x ] of polynomials over R in the commutative indeterminate x . For each radical property S we define the function S * which assigns to each ring R the ideal of R . It is shown that the property S A (that a ring R be equal to S * ( R )) is a radical property. If S is semiprime, then S A is semiprime also. If S is a special radical, then S A is a special radical. S A is always contained in S . A necessary and sufficient condition that S and S A coincide is given. The results are generalized in the last section to include extensions of R other than R [ x ], One such extension is the semigroup ring R [ A ], where A is a semigroup with an identity adjoined.