A Construction in General Radical Theory
1970 ◽
Vol 22
(6)
◽
pp. 1097-1100
◽
Keyword(s):
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.
1990 ◽
Vol 32
(2)
◽
pp. 180-192
◽
1990 ◽
Vol 42
(2)
◽
pp. 315-341
◽
1979 ◽
Vol 31
(2)
◽
pp. 255-263
◽
1978 ◽
Vol 26
(1)
◽
pp. 31-45
◽
1972 ◽
Vol 18
(2)
◽
pp. 129-136
◽
1981 ◽
Vol 89
(1-2)
◽
pp. 25-50
◽
1996 ◽
Vol 39
(3)
◽
pp. 275-283
◽
1963 ◽
Vol 6
(2)
◽
pp. 267-273
◽
1982 ◽
Vol 34
(3)
◽
pp. 718-736
◽
1981 ◽
Vol 91
(1-2)
◽
pp. 135-145