Using Survival Pairs to Characterize Rings of Algebraic Integers
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let $R$ be a domain with quotient field $K$ and prime subring $A$. Then $R$ is integral over each of its subrings having quotient field $K$ if and only if $(A,R)$ is a survival pair. This shows the redundancy of a condition involving going-down pairs in a earlier characterization of such rings. In characteristic $0$, the domains being characterized are the rings $R$ that are isomorphic to subrings of the ring of all algebraic integers. In positive (prime) characteristic, the domains $R$ being characterized are of two kinds: either $R = K$ is an algebraic field extension of $A$ or precisely one valuation domain of $K$ does not contain $R$.
Abstract
Let $R$ be a domain with quotient field $K$ and prime subring $A$. Then $R$ is integral over each of its subrings having quotient field $K$ if and only if $(A,R)$ is a survival pair. This shows the redundancy of a condition involving going-down pairs in a earlier characterization of such rings. In characteristic $0$, the domains being characterized are the rings $R$ that are isomorphic to subrings of the ring of all algebraic integers. In positive (prime) characteristic, the domains $R$ being characterized are of two kinds: either $R = K$ is an algebraic field extension of $A$ or precisely one valuation domain of $K$ does not contain $R$.
참고문헌 (15)
- A. Ayache, "Universally catenarian and going-down pairs of rings", Math. Z., 238, pp.695~731 (2001)
- A. Ayache, "Residually algebraic pairs of rings", Math. Z., 225, pp.49~65 (1997)
- N. Bourbaki, "Commutative algebra" (1972), Addison-Wesley Publishing Co.
- R. D. Chatham, "Going-down pairs of commutative rings", Rend. Circ. Mat. Palermo (2), 50, pp.509~542 (2001)
- J. Coykendall, "Survival pairs of commutative rings have the lyingover property", Comm. Algebra, 31, pp.259~270 (2003)
- J. Coykendall, "A generalization of integrality", Canad. Math. Bull, 53, pp.639~653 (2010)
- D. E. Dobbs, "On INC-extensions and polynomials with unit content", Canad. Math. Bull., 23, pp.37~42 (1980)
- D. E. Dobbs, "Lying-over pairs of commutative rings", Canad. J. Math., 33, pp.454~475 (1981)
- D. E. Dobbs, "Going-down underrings", Bull. Austral. Math. Soc., 36, pp.503~513 (1987)
- D. E. Dobbs, "Domains integral over each underring", Tokyo J. Math, 10, pp.317~325 (1987)
- R. Gilmer, "Integral domains with Noetherian subrings", Comment. Math. Helv., 45, pp.129~134 (1970)
- R. Gilmer, "Multiplicative ideal theory" (1972), Marcel Dekker
- I. Kaplansky, "Commutative rings" (1974), The University of Chicago Press
- F. Richman, "Generalized quotient rings", Proc. Amer. Math. Soc., 16, pp.794~799 (1965)
- O. Zariski, "Commutative algebra, Vol I" (1958), Van Nostrand Company