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 idealof R. It is shown that the property SA (that a ring R be equal to S*(R)) is a radical property. If S is semiprime, then SA is semiprime also. If S is a special radical, then SA is a special radical. SA is always contained in S. A necessary and sufficient condition that S and SA 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.