Normal Pairs of Going-down Rings
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Let (R; T) be a normal pair of commutative rings (i.e., R T is a unital extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R S T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the \if" part of the assertion fails if P is taken to be the \divided domain" property.
Abstract
Let (R; T) be a normal pair of commutative rings (i.e., R T is a unital extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R S T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the \if" part of the assertion fails if P is taken to be the \divided domain" property.
참고문헌 (24)
- A. Badawi, "On divided commutative rings", Comm. Algebra, 27, pp.1465~1474 (1999)
- A. Badawi, "On locally divided rings and going-down rings", Comm. Algebra, 29, pp.2805~2825 (2001)
- J. Coykendall, "Saturated chains of integrally closed overrings", JP J. Algebra, Number Theory Appl, 13, pp.121~130 (2009)
- E. D. Davis, "Overrings of commutative rings. III: normal pairs", Trans. Amer. Math. Soc, 182, pp.175~185 (1973)
- D. E. Dobbs, "On going-down for simple overrings, II", Comm. Algebra, 1, pp.439~458 (1974)
- D. E. Dobbs, "Divided rings and going-down", Pac. J. Math, 67, pp.353~363 (1976)
- D. E. Dobbs, "On Henselian pullbacks", Lecture Notes Pure Appl. Math, 189, pp.317~326 (1997)
- D. E. Dobbs, "Going-down rings with zero-divisors", Houston J. Math, 23, pp.1~12 (1997)
- D. E. Dobbs, "When is a pullback a locally divided domain", Houston J. Math, 35, pp.341~351 (2009)
- D. E. Dobbs, "When a minimal overring is a going-down domain", Houston J. Math, 36, pp.33~42 (2010)
- D. E. Dobbs, "On going-down for simple overrings, III", Proc. Amer. Math. Soc, 54, pp.147~168 (1976)
- D. E. Dobbs, "Weak Baer going-down rings", Houston J. Math, 29, pp.559~581 (2003)
- D. E. Dobbs, "A generalization of divided domains and its connection to weak Baer going-down rings", Comm. Algebra, 37, pp.3553~3572 (2009)
- D. E. Dobbs, "INC-extensions amid zero-divisors", Int. Electron. J. Algebra, 7, pp.102~109 (2010)
- D. E. Dobbs, "Normal pairs with zero-divisors", J. Algebra Appl
- D. Ferrand, "Morphismes minimaux", J. Algebra, 16, pp.461~471 (1970)
- M. Fontana, "Topologically de ned classes of commutative rings", Ann. Mat. Pura Appl, 123, pp.331~355 (1980)
- R. Gilmer, "Multiplicative Ideal Theory" (1972), Dekker
- M. Griffin, "Prufer rings with zero divisors", J. Reine Angew. Math, 240, pp.55~67 (1970)
- J. A. Huckaba, "Commutative Rings with Zero Divisors" (1988), Dekker
- I. Kaplansky, "Commutative Rings" (1974), Univ. Chicago Press
- S. McAdam, "Simple going down", J. London Math. Soc, 13, pp.167~173 (1976)
- M. Nagata, "Local Rings" (1962), Wiley-Interscience
- J. Sato, "On minimal overrings of a Noetherian domain", Comm. Algebra, 20, pp.1746~1753 (1992)