scholarly journals Hereditary Local Rings

1966 ◽  
Vol 27 (1) ◽  
pp. 223-230 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Local Ring ◽  
Local Rings ◽  
Unique Factorization ◽  
Prime Factorization ◽  
Unique Factorization Domain ◽  
Necessary And Sufficient

Many questions about free ideal rings ( = firs, cf. [5] and §2 below) which at present seem difficult become much easier when one restricts attention to local rings. One is then dealing with hereditary local rings, and any such ring is in fact a fir (§2). Our object thus is to describe hereditary local rings. The results on firs in [5] show that such a ring must be a unique factorization domain; in §3 we prove that it must also be rigid (cf. the definition in [3] and §3 below). More precisely, for a semifir R with prime factorization rigidity is necessary and sufficient for R to be a local ring.

scholarly journals VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY

2018 ◽  
Vol 61 (03) ◽  
pp. 705-725
Author(s):  
DIPANKAR GHOSH ◽  
TONY J. PUTHENPURAKAL
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Regular Sequence ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Syzygy Modules ◽  
Tor Modules ◽  
Necessary And Sufficient ◽  
Minimal Multiplicity

AbstractLet R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.

An application of local uniformization to the theory of divisors

1951 ◽  
Vol 47 (2) ◽  
pp. 279-285
Author(s):  
D. G. Northcott
Keyword(s):  
Quotient Ring ◽  
Local Rings ◽  
Simple Point ◽  
Unique Factorization ◽  
The Third ◽  
Unique Factorization Domain ◽  
Irreducible Variety ◽  
Structure Theorems ◽  
Local Uniformization

If V is an irreducible variety and W is an irreducible simple subvariety of V, then one of the properties of the quotient ring of W in V is that it is a unique factorization domain. A proof of this theorem has been given by Zariski ((2), Theorem 5, p. 22), based on the structure theorems for complete local rings, and the fact that the local rings which arise geometrically are always analytically unramified. Here the theorem is deduced from certain properties of functions and their divisors which will be established by entirely different considerations. The terminology which will be employed is that proposed by A. Weil in his book(1), and we shall use, for instance, F-viii, Th. 3, Cor. 1, when referring to Corollary 1 of the third theorem in Chapter 8. Before proceeding to details it should be noted that Weil and Zariski differ in then-definitions, and that in particular the terms ‘variety’ and ‘simple point’ do not mean quite the same in the two theories. The effect of this is to make Zariski's result somewhat stronger than Theorem 3 of this paper.

Artinian Local Rings Whose Annihilating-ideal Graphs Are Star Graphs

2015 ◽  
Vol 22 (01) ◽  
pp. 73-82 ◽  
Author(s):  
Houyi Yu ◽  
Tongsuo Wu ◽  
Weiping Gu
Keyword(s):  
Local Ring ◽  
Complete Characterization ◽  
Local Rings ◽  
Star Graph ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Star Graphs ◽  
Finite Local Ring ◽  
Necessary And Sufficient

In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete characterization is established for a finite local ring whose annihilating-ideal graph is a star graph.

Formal Fibers of Unique Factorization Domains

2010 ◽  
Vol 62 (4) ◽  
pp. 721-736 ◽  
Author(s):  
Adam Boocher ◽  
Michael Daub ◽  
Ryan K. Johnson ◽  
H. Lindo ◽  
S. Loepp ◽  
...  
Keyword(s):  
Noetherian Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Unique Factorization ◽  
Tight Closure ◽  
Unique Factorization Domain ◽  
Necessary And Sufficient ◽  
Unique Factorization Domains ◽  
Adic Completion

AbstractLet (T,M) be a complete local (Noetherian) ring such that dimT ≥ 2 and |T| = |T/M| and let ﹛pi﹜i∈𝒥 be a collection of elements of T indexed by a set I so that |𝒥| < |T|. For each i ∈ 𝒥, let Ci := ﹛Qi1, … ,Qini ﹜ be a set of nonmaximal prime ideals containing pi such that the Qi j are incomparable and pi ∈ Qjk if and only if i = j. We provide necessary and sufficient conditions so that T is the m -adic completion of a local unique factorization domain (A,m ), and for each i ∈ I, there exists a unit ti of T so that pi ti ∈ A andCi is the set of prime ideals Q of T that are maximal with respect to the condition that Q ∩ A = pi tiA.We then use this result to construct a (nonexcellent) unique factorization domain containingmany ideals for which tight closure and completion do not commute. As another application, we construct a unique factorization domain A most of whose formal fibers are geometrically regular.

scholarly journals Some criteria for regular and Gorenstein local rings via syzygy modules

2019 ◽  
Vol 18 (05) ◽  
pp. 1950097
Author(s):  
Dipankar Ghosh
Keyword(s):  
Direct Summand ◽  
Local Ring ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Local Rings ◽  
Canonical Module ◽  
Syzygy Module ◽  
Syzygy Modules ◽  
Necessary And Sufficient ◽  
Minimal Multiplicity

Let [Formula: see text] be a Cohen–Macaulay local ring. We prove that the [Formula: see text]th syzygy module of a maximal Cohen–Macaulay [Formula: see text]-module cannot have a semidualizing direct summand for every [Formula: see text]. In particular, it follows that [Formula: see text] is Gorenstein if and only if some syzygy of a canonical module of [Formula: see text] has a nonzero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen–Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.

Unique Factorization Theorems for Subalgebras of the Incidence Algebra

1972 ◽  
Vol 24 (5) ◽  
pp. 967-977
Author(s):  
K. L. Yocom
Keyword(s):  
Partially Ordered Set ◽  
Integral Domain ◽  
Ordered Set ◽  
Sufficient Conditions ◽  
Unique Factorization ◽  
Countable Number ◽  
Incidence Algebra ◽  
Direct Sum ◽  
Unique Factorization Domain ◽  
Necessary And Sufficient

H. Scheid [4] has found necessary and sufficient conditions on a partially ordered set S(≦) which is a direct sum of a countable number of trees for a certain subalgebra G(+, *) of the incidence algebra F(+, *) to be an integral domain. In this paper we prove that under similar conditions on S, G(+, *) is actually a unique factorization domain or, failing this, that there is a subalgebra H(+, *) of F(+, *) which is a unique factorization domain and contains G. Similar results are then obtained as corollaries in the regular convolution rings of Narkiewicz.

scholarly journals Simple Deformed Witt Algebras

2011 ◽  
Vol 18 (03) ◽  
pp. 533-540 ◽  
Author(s):  
Guang'ai Song ◽  
Chunguang Xia
Keyword(s):  
Associative Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Unique Factorization ◽  
Witt Algebra ◽  
Unique Factorization Domain ◽  
Necessary And Sufficient ◽  
Algebra Endomorphism

For any unique factorization domain [Formula: see text] and an algebra endomorphism σ of [Formula: see text], there exists a non-associative algebra [Formula: see text] with multiplication satisfying skew-symmetry and generalized (twisted) Jacobi identities, called a σ-deformed Witt algebra. In this paper, we obtain necessary and sufficient conditions for the algebra [Formula: see text] to be simple.

Two-Dimensional Linear Groups Over Local Rings

1969 ◽  
Vol 21 ◽  
pp. 106-135 ◽  
Author(s):  
Norbert H. J. Lacroix
Keyword(s):  
Local Ring ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Linear Groups ◽  
Local Rings ◽  
Two Dimensional ◽  
Normal Subgroups ◽  
Field Case

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .

Reduction numbers and Hilbert polynomials of ideals in higher dimensional CohenMacaulay local rings

1992 ◽  
Vol 111 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Yinghwa Wu
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Primary Ideal ◽  
Local Rings ◽  
Hilbert Polynomials ◽  
Higher Dimensional ◽  
Reduction Number ◽  
Minimal Reduction ◽  
System Of Parameters

Throughout, (R, m) will denote a d-dimensional CohenMacaulay (CM for short) local ring having an infinite residue field and I an m-primary ideal in R. Recall that an ideal J I is said to be a reduction of I if Ir+1 = JIr for some r 0, and a reduction J of I is called a minimal reduction of I if J is generated by a system of parameters. The concepts of reduction and minimal reduction were first introduced by Northcott and Rees12. If J is a reduction of I, define the reduction number of I with respect to J, denoted by rj(I), to be min {r 0 Ir+1 = JIr}. The reduction number of I is defined as r(I) = min {rj(I)J is a minimal reduction of I}. The reduction number r(I) is said to be independent if r(I) = rj(I) for every minimal reduction J of I.

scholarly journals Linearity defect of the residue field of short local rings

2016 ◽  
Vol 16 (09) ◽  
pp. 1750163
Author(s):  
Rasoul Ahangari Maleki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Residue Field ◽  
Positive Answer ◽  
Local Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Linear Resolution

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text] and residue field [Formula: see text]. The linearity defect of a finitely generated [Formula: see text]-module [Formula: see text], which is denoted [Formula: see text], is a numerical measure of how far [Formula: see text] is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether [Formula: see text] implies [Formula: see text], in the case when [Formula: see text].

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