scholarly journals The canonical modules of graded rings defined by generic matrices

1981 ◽  
Vol 81 ◽  
pp. 105-112 ◽  
Author(s):  
Yuji Yoshino
Keyword(s):  
Polynomial Ring ◽  
Quotient Ring ◽  
Graded Rings ◽  
Canonical Module ◽  
Graded Ring ◽  
The Ideal

Let k be a field, and X = [xij] be an n × (n + m) matrix whose elements are algebraically independent over k.We shall study the canonical module of the graded ring R, which is a quotient ring of the polynomial ring A = k[X] by the ideal αn(X) generated by all the n × n minors of X.

scholarly journals Behavior of the Auslander condition with respect to regradings

2019 ◽  
Vol 18 (09) ◽  
pp. 1950168 ◽  
Author(s):  
G.-S. Zhou ◽  
Y. Shen ◽  
D.-M. Lu
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Noetherian Ring ◽  
Invertible Element ◽  
Quotient Ring ◽  
Global Dimension ◽  
Injective Dimension ◽  
Graded Ring ◽  
Ideal Formula ◽  
The Ideal

We show that a noetherian ring graded by an abelian group of finite rank satisfies the Auslander condition if and only if it satisfies the graded Auslander condition. In addition, we also study the injective dimension, the global dimension and the Cohen–Macaulay property from the same perspective as that for the Auslander condition. A key step of our approach is to establish homological relations between a graded ring [Formula: see text], its quotient ring modulo the ideal [Formula: see text] and its localization ring with respect to the Ore set [Formula: see text], where [Formula: see text] is a homogeneous regular normal non-invertible element of [Formula: see text].

On the equations defining tangent cones

1980 ◽  
Vol 88 (2) ◽  
pp. 281-297 ◽  
Author(s):  
Lorenzo Robbiano ◽  
Giuseppe Valla
Keyword(s):  
Local Ring ◽  
Tangent Cones ◽  
Homogeneous Ideal ◽  
Graded Rings ◽  
Graded Ring ◽  
Local Study ◽  
Basic Facts ◽  
Image Position ◽  
Initial Forms ◽  
The Ideal

This paper treats the local study of singularities by means of their tangent cones, more specifically the study of graded rings associated to an ideal of a local ring. We recall some basic facts: let (R,) be a local ring,I, Jideals ofR, such thatJ⊆I; thenGR/J(I/J), the graded ring associated toI/J, is canonically isomorphic to the quotient ofGR(I) modulo a homogeneous ideal, which is calledJ*, and which is generated by the so-called ‘initial forms’ of the elements ofJ. Let us consider the following example: Letkbe a field,R=k[X, Y, Z](x, y, Z),I= (X, Y, Z)R, Jthe prime ideal generated byfl,f2wheref1=Y3−Z2,f2=YZ−X4. Thenand it is easily seen thatJ*properly contains the ideal generated by the initial formsf*1f*2off1,f2; namelyf*1= −Z2,f*2=YZand (Yf1+Zf2)* =Y4∉ (−Z2,YZ).

Strongly graded rings which are generalized Dedekind rings

2019 ◽  
Vol 19 (03) ◽  
pp. 2050043
Author(s):  
Sri Wahyuni ◽  
Hidetoshi Marubayashi ◽  
Iwan Ernanto ◽  
Sutopo
Keyword(s):  
Quotient Ring ◽  
Dedekind Ring ◽  
Graded Rings ◽  
Graded Ring ◽  
Type Formula

Let [Formula: see text] be a strongly graded ring of type [Formula: see text] such that [Formula: see text] is a prime Goldie ring with its quotient ring [Formula: see text]. It is shown that the following three conditions are equivalent: (i) [Formula: see text] is a [Formula: see text]-invariant generalized Dedekind ring ([Formula: see text]-Dedekind ring for short), (ii) [Formula: see text] is a [Formula: see text]-Dedekind ring and (iii) [Formula: see text] is a graded [Formula: see text]-Dedekind ring. We describe all invertible ideals of [Formula: see text]-Dedekind rings in terms of [Formula: see text] and [Formula: see text]. We provide counterexamples of [Formula: see text]-invariant [Formula: see text]-Dedekind rings which are not [Formula: see text]-Dedekind rings.

scholarly journals The close relation between border and Pommaret marked bases

2021 ◽  
Author(s):  
Cristina Bertone ◽  
Francesca Cioffi
Keyword(s):  
Finite Order ◽  
Polynomial Ring ◽  
Group Actions ◽  
Close Relation ◽  
Open Covering ◽  
Order Ideal ◽  
Lower Dimension ◽  
Hilbert Schemes ◽  
Affine Spaces ◽  
The Ideal

AbstractGiven a finite order ideal $${\mathcal {O}}$$ O in the polynomial ring $$K[x_1,\ldots , x_n]$$ K [ x 1 , … , x n ] over a field K, let $$\partial {\mathcal {O}}$$ ∂ O be the border of $${\mathcal {O}}$$ O and $${\mathcal {P}}_{\mathcal {O}}$$ P O the Pommaret basis of the ideal generated by the terms outside $${\mathcal {O}}$$ O . In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $$\partial {\mathcal {O}}$$ ∂ O -marked sets (resp. bases) and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked sets (resp. bases). We prove that a $$\partial {\mathcal {O}}$$ ∂ O -marked set B is a marked basis if and only if the $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $$\partial {\mathcal {O}}$$ ∂ O -marked bases and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.

scholarly journals A NOTE ON NIL AND JACOBSON RADICALS IN GRADED RINGS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350121 ◽  
Author(s):  
AGATA SMOKTUNOWICZ
Keyword(s):  
Jacobson Radical ◽  
Analogous Result ◽  
Graded Rings ◽  
Radical Ring ◽  
Graded Ring ◽  
Nil Ring ◽  
Nil Radical

It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil radicals, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper, it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some new results on graded-nil rings are also obtained.

On Strongly Groupoid Graded Rings and the Corresponding Clifford Theorem

2006 ◽  
Vol 13 (02) ◽  
pp. 181-196 ◽  
Author(s):  
Gongxiang Liu ◽  
Fang Li
Keyword(s):  
Main Task ◽  
Graded Rings ◽  
Graded Ring ◽  
Matrix Rings ◽  
Definition Of

In this paper, we introduce the definition of groupoid graded rings. Group graded rings, (skew) groupoid rings, artinian semisimple rings, matrix rings and others can be regarded as special kinds of groupoid graded rings. Our main task is to classify strongly groupoid graded rings by cohomology of groupoids. Some classical results about group graded rings are generalized to groupoid graded rings. In particular, the Clifford Theorem for a strongly groupoid graded ring is given.

scholarly journals When Are Graded Rings Graded S-Noetherian Rings

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1532
Author(s):  
Dong Kyu Kim ◽  
Jung Wook Lim
Keyword(s):  
Noetherian Ring ◽  
Semigroup Ring ◽  
Noetherian Rings ◽  
Homogeneous Ideal ◽  
Commutative Monoid ◽  
Graded Rings ◽  
Graded Ring ◽  
Special Case

Let Γ be a commutative monoid, R=⨁α∈ΓRα a Γ-graded ring and S a multiplicative subset of R0. We define R to be a graded S-Noetherian ring if every homogeneous ideal of R is S-finite. In this paper, we characterize when the ring R is a graded S-Noetherian ring. As a special case, we also determine when the semigroup ring is a graded S-Noetherian ring. Finally, we give an example of a graded S-Noetherian ring which is not an S-Noetherian ring.

scholarly journals Group gradations on Leavitt path algebras

2019 ◽  
Vol 19 (09) ◽  
pp. 2050165 ◽  
Author(s):  
Patrik Nystedt ◽  
Johan Öinert
Keyword(s):  
Directed Graph ◽  
Path Algebra ◽  
Graded Rings ◽  
Leavitt Path Algebra ◽  
Arbitrary Group ◽  
Graded Ring ◽  
Identity Component ◽  
Path Algebras ◽  
Leavitt Path Algebras

Given a directed graph [Formula: see text] and an associative unital ring [Formula: see text] one may define the Leavitt path algebra with coefficients in [Formula: see text], denoted by [Formula: see text]. For an arbitrary group [Formula: see text], [Formula: see text] can be viewed as a [Formula: see text]-graded ring. In this paper, we show that [Formula: see text] is always nearly epsilon-strongly [Formula: see text]-graded. We also show that if [Formula: see text] is finite, then [Formula: see text] is epsilon-strongly [Formula: see text]-graded. We present a new proof of Hazrat’s characterization of strongly [Formula: see text]-graded Leavitt path algebras, when [Formula: see text] is finite. Moreover, if [Formula: see text] is row-finite and has no source, then we show that [Formula: see text] is strongly [Formula: see text]-graded if and only if [Formula: see text] has no sink. We also use a result concerning Frobenius epsilon-strongly [Formula: see text]-graded rings, where [Formula: see text] is finite, to obtain criteria which ensure that [Formula: see text] is Frobenius over its identity component.

scholarly journals Chain conditions and semigroup graded rings

1988 ◽  
Vol 45 (3) ◽  
pp. 372-380 ◽  
Author(s):  
E. Jespers
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Cancellative Semigroup ◽  
Graded Rings ◽  
Semigroup Rings ◽  
Graded Ring ◽  
Product Group ◽  
Chain Conditions ◽  
Unique Product

AbstractThe following questions are studied: When is a semigroup graded ring left Noetherian, respectively semiprime left Goldie? Necessary sufficient conditions are proved for cancellative semigroup-graded subrings of rings weakly or strongly graded by a polycyclic-by-finite (unique product) group. For semigroup rings R[S] we also give a solution to the problem in case S is an inverse semigroup.

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