The Factor Domains that Result from Uppers to Prime Ideals in Polynomial Rings
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Let P be a prime ideal of a commutative unital ring R; X an indeterminate;D := R/P; L the quotient field of D; F an algebraic closure of L; alpha∈ L[X] a monic irreducible polynomial; xi any root of alpha in F; and Q = <P,alpha>, the upper to P with respect to alpha. Then R[X]/Q is R-algebra isomorphic to D[xi]; and is R-isomorphic to an overring of D if and only if deg(alpha) = 1.
Abstract
Let P be a prime ideal of a commutative unital ring R; X an indeterminate;D := R/P; L the quotient field of D; F an algebraic closure of L; alpha∈ L[X] a monic irreducible polynomial; xi any root of alpha in F; and Q = <P,alpha>, the upper to P with respect to alpha. Then R[X]/Q is R-algebra isomorphic to D[xi]; and is R-isomorphic to an overring of D if and only if deg(alpha) = 1.
참고문헌 (8)
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