THE KRONECKER FUNCTION RING OF THE RING D[X]N
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Let D be an integrally closed domain with quotient field K,* be a star operation on D, X, Y be indeterminates over D, N* = {f ∈D[X]| (cD(f))* = D} and R = D[X]N* . Let b be the b-operation on R,and let *c be the star operation on D defined by I*c = (ID[X]N* )b ∩K. Finally, let Kr(R, b) (resp., Kr(D, *c)) be the Kronecker function ring of R (resp., D) with respect to Y (resp., X, Y). In this paper, we show that Kr(R, b) ⊆ Kr(D, *c) and Kr(R, b) is a kfr with respect to K(Y ) and X in the notion of [2]. We also prove that Kr(R, b) = Kr(D, *c) if and only if D is a P*MD. As a corollary, we have that if D is not a P*MD, then Kr(R, b) is an example of a kfr with respect to K(Y) and X but not a Kronecker function ring with respect to K(Y ) and X.
Abstract
Let D be an integrally closed domain with quotient field K,* be a star operation on D, X, Y be indeterminates over D, N* = {f ∈D[X]| (cD(f))* = D} and R = D[X]N* . Let b be the b-operation on R,and let *c be the star operation on D defined by I*c = (ID[X]N* )b ∩K. Finally, let Kr(R, b) (resp., Kr(D, *c)) be the Kronecker function ring of R (resp., D) with respect to Y (resp., X, Y). In this paper, we show that Kr(R, b) ⊆ Kr(D, *c) and Kr(R, b) is a kfr with respect to K(Y ) and X in the notion of [2]. We also prove that Kr(R, b) = Kr(D, *c) if and only if D is a P*MD. As a corollary, we have that if D is not a P*MD, then Kr(R, b) is an example of a kfr with respect to K(Y) and X but not a Kronecker function ring with respect to K(Y ) and X.
참고문헌 (10)
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