Universal Lying-Over Rings |
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| Authors: | David E. Dobbs Jay Shapiro |
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| Affiliation: | 1. Department of Mathematics , University of Tennessee , Knoxville, Tennessee, USA dobbs@math.utk.edu;3. Department of Mathematics , George Mason University , Fairfax, Virginia, USA |
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| Abstract: | A (commutative unital) ring R is said to satisfy universal lying-over (ULO) if each injective ring homomorphism R → T satisfies the lying-over property. If R satisfies ULO, then R = tq(R), the total quotient ring of R. If a reduced ring satisfies ULO, it also satisfies Property A. If a ring R = tq(R) satisfies Property A and each nonminimal prime ideal of R is an intersection of maximal ideals, R satisfies ULO. If 0 ≤ n ≤ ∞, there exists a reduced (resp., nonreduced) n-dimensional ring satisfying ULO. The A + B construction is used to show that if 2 ≤ n < ∞, there exists an n-dimensional reduced ring R such that R = tq(R), R satisfies Property A, but R does not satisfy ULO. |
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| Keywords: | A + B construction Hilbert ring h-local domain Krull dimension Lying-over Property A Reduced ring Total quotient ring |
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