THE S-FINITENESS ON QUOTIENT RINGS OF A POLYNOMIAL RING

초록

Let R be a commutative ring with identity, R[X] the polynomial ring over R and S a multiplicative subset of R. Let U = {f is an element of R[X] vertical bar f is monic} and let N = {f is an element of R[X] vertical bar c(f) = R}. In this paper, we show that if S is an anti-Archimedean subset of R, then R is an S-Noetherian ring if and only if R[X] U is an S-Noetherian ring, if and only if R[X] N is an S-Noetherian ring. We also prove that if R is an integral domain and R[X] U is an S-principal ideal domain, then R is an S-principal ideal domain.

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