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 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.