Prüfer and valuation rings with zero divisors

Boisen, M. B.; Larsen, Max D.

doi:10.2140/pjm.1972.40.7

Cited by 17 publications

(12 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…is a valuation pair for all regulδr maximal ideals M of R. In [1] it is shown that Prόfer rings satisfy property C. It is easy to see that valuation rings are integrally closed. Griffin shows in [3] that the following Statements are equivalent :…”

Section: [M]r [M] )mentioning

confidence: 99%

The containment property for large quotient rings.

1973

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

Section: [M]r [M] )mentioning

confidence: 99%

The containment property for large quotient rings.

1973

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

“…Since R is reduced, P Π Q = (0) and thus R can be embedded in R/P θ R/Q. Moreover, R/P θ R/Q is integral over R since both (1,0) and (0,1) satisfy the polynomial equation X 2 -X = 0. Thus if both R/P and R/Q are Prufer domains, then the integral closure of R in T(R/P θ R/Q) is exactly R/P θ R/Q and hence is arithmetical.…”

Section: Lemma 25 Let R Be a Reduced Ring With Exactly Two Minimal mentioning

confidence: 99%

Some results on Prüfer rings

Lucas¹

1986

Pacific J. Math.

View full text Add to dashboard Cite

For a domain D, D is a Prufer domain if and only if D/P is aPrufer domain for every prime ideal P of D. The same result does not hold for rings with zero divisors. In this paper it is shown that for a Prufer ring R with prime ideal P, R/P is a Prufer ring if P is not properly contained in an ideal consisting entirely of zero divisors. An example is provided to show that, in general, this is the best possible result. According to M. Boisen and P. Sheldon, a pre-Prufer ring is defined to be a ring for which every proper homomorphic image is a Prufer ring. In this paper it is proved that for a pre-Prufer ring R containing zero divisors, the integral closure of R is a Prufer ring. Furthermore, if R is a reduced pre-Prufer ring with more than two minimal prime ideals, then R is already integrally closed and, moreover, R is not only Prufer but arithmetical as well. An example is provided of an integrally closed pre-Prufer domain which is not a Prufer domain.1. Introduction. In this paper, all rings are assumed to be commutative with nonzero unit. A regular element of a ring R is one which is not a divisor of zero, and a regular ideal is one which contains a regular element. By an overling we mean a ring between R and T(R), the total quotient ring of R. All unexplained terminology is standard as in [5] and [8].For commutative rings with zero divisors, M. Griffin [6] defined a Prufer ring as a ring for which every finitely generated regular ideal is invertible. He showed that with suitable modifications many of the properties which characterize Prufer domains also characterize Prufer rings; some of these are collected in Proposition 2.1 of this paper. One purpose of this paper is to examine what can be said about the relation between a ring R being Prufer and R/P being a Prufer domain for each prime ideal P of R. It is not the case that the two conditions are equivalent. In fact, in general neither implies the other. In §2 we provide the following positive results. First, if P is either a regular prime ideal or maximal with respect to containing only zero divisors, then R/P is a Prufer ring if R is a Prufer ring (Proposition 2.2). In the event that T(R) is von Neumann regular, then every prime ideal of R satisfies the above condition. Thus if R is Prufer and T(R) is von Neumann regular, then R/P is a Prufer domain for every prime ideal P of R. Moreover, if R/P

show abstract

“…Finally, we say that a ring A with the total quotient ring A" is a ^-N-ring for some ring topology (3) => (1). Let m be a maximal regular ideal of Â and P = m n A.…”

mentioning

confidence: 99%