On the Integral Degree of Integral Ring Extensions (original) (raw)
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On the complete integral closure of an integral domain
Journal of the Australian Mathematical Society, 1966
We consider in this paper only commutative rings with identity. When R is considered as a subring of S it will always be assumed that R and 5 have the same identity. If R is a subring of 5 an element s of 5 is said to be integral over R if s is the root of a monic polynomial with coefficients in R. Following Krull [8], p. 102, we say s is almost integral over R provided all powers of s belong to a finite i?-submodule of S. * If R x is the set of elements of S almost integral over R we say R 1 is the complete integral closure of R in S. If R = i? 2 we say R is completely integrally closed in S. If R x = S we say S is almost integral over R. If S is the total quotient ring of R, we call R ± the complete integral closure of R, and in this case if R = R x we say simply R is completely integrally closed. The terms integral closure of R in S, R is integrally closed in S, S is integral over Ri the integral closure of R, and R is integrally closed are similarly defined. For elementary results on these properties, see , Ch. 14, and [15], Ch. 5. This paper is principally concerned with the complete integral closure D* of an integral domain D and a determination of when D* is completely integrally closed, though some results in more general cases are obtained.
On some classes of integral domains defined by Krull's a.b. operations
2011
Let D be an integral domain with quotient field K. The b-operation that associates to each nonzero D-submodule E of K, E^b := {EV | V valuation overring of D}, is a semistar operation that plays an important role in many questions of ring theory (e.g., if I is a nonzero ideal in D, I^b coincides with its integral closure). In a first part of the paper, we study the integral domains that are b-Noetherian (i.e., such that, for each nonzero ideal I of D, I^b = J^b for some a finitely generated ideal J of D). For instance, we prove that a b-Noetherian domain has Noetherian spectrum and, if it is integrally closed, is a Mori domain, but integrally closed Mori domains with Noetherian spectra are not necessarily b-Noetherian. We also characterize several distinguished classes of b-Noetherian domains. In a second part of the paper, we study more generally the e.a.b. semistar operation of finite type _a canonically associated to a given semistar operation (for instance, the b-operation is th...
Some properties of integral closure
Proceedings of the American Mathematical Society, 1967
Let D be an integrally closed domain with identity having quotient field K, let L be an algebraic extension field of K, and let D be the integral closure of D in L. We prove here that the following five ideal theoretic structure properties of D are inherited by D, namely: (a) D is a Priifer domain, (b) D is an almost Dedekind domain,1 (c) D is a Dedekind domain, (d) D has the QA-property,2 (e) D has property (#).3
Matlis’ semi-regularity in trivial ring extensions of integral domains
Colloquium Mathematicum
This paper contributes to the study of homological aspects of trivial ring extensions (also called Nagata idealizations). Namely, we investigate the transfer of the notion of (Matlis') semi-regular ring (also known as IF-ring) along with related concepts, such as coherence, in trivial ring extensions issued from integral domains. All along the paper, we put the new results in use to enrich the literature with new families of examples subject to semi-regularity.
Codimension, multiplicity and integral extensions
Mathematical Proceedings of the Cambridge Philosophical Society, 2001
Let A ⊂ B be a homogeneous inclusion of standard graded algebras with A 0 = B 0 . To relate properties of A and B we intermediate with another algebra, the associated graded ring G = gr A1B (B). We give criteria as to when the extension A ⊂ B is integral or birational in terms of the codimension of certain modules associated to G. We also introduce a series of multiplicities associated to the extension A ⊂ B. There are applications to the extension of two Rees algebras of modules and to estimating the (ordinary) multiplicity of A in terms of that of B and of related rings. Many earlier results by several authors are recovered quickly.
Mathematics of the USSR-Sbornik, 1982
This paper studies integrally closed rings. It is shown that a semiprime integrally closed Goldie ring is the direct product of a semisimple artinian ring and a finite number of integrally closed invariant domains that are classically integrally closed in their (division) rings of fractions. It is shown also that an integrally closed ring has a classical ring of fractions and is classically integrally closed in it.
On some classes of integral domains defined by Krullʼs a.b. operations
Journal of Algebra, 2011
Let D be an integral domain with quotient field K. The boperation that associates to each nonzero D-submodule E of K, E b := {EV | V valuation overring of D}, is a semistar operation that plays an important role in many questions of ring theory (e.g., if I is a nonzero ideal in D, I b coincides with its integral closure). In a first part of the paper, we study the integral domains that are b-Noetherian (i.e., such that, for each nonzero ideal I of D, I b = J b for some a finitely generated ideal J of D). For instance, we prove that a b-Noetherian domain has Noetherian spectrum and, if it is integrally closed, is a Mori domain, but integrally closed Mori domains with Noetherian spectra are not necessarily b-Noetherian. We also characterize several distinguished classes of b-Noetherian domains. In a second part of the paper, we study more generally the e.a.b. semistar operation of finite type ⋆a canonically associated to a given semistar operation ⋆ (for instance, the b-operation is the e.a.b. semistar operation of finite type canonically associated to the identity operation). These operations, introduced and studied by Krull, Jaffard, Gilmer and Halter-Koch, play a very important role in the recent generalizations of the Kronecker function ring. In particular, in the present paper, we classify several classes of integral domains having some of the fundamental operations d, t, w and v equal to some of the canonically associated e.a.b. operations b, ta, wa and va.