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.

Multiplicities in graded rings II: integral equivalence and the Buchsbaum–Rim multiplicity

1996 ◽  
Vol 119 (3) ◽  
pp. 425-445 ◽  
Author(s):  
D. Kirby ◽  
D. Rees
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Noetherian Ring ◽  
Free Module ◽  
Graded Rings ◽  
Finitely Generated ◽  
Graded Ring ◽  
The Subject ◽  
Integral Equivalence ◽  
Image Position

While this paper is principally a continuation of [5], with as its object the application of sections 6 and 7 of that paper to obtain results related to the Buchsbaum–Rim multiplicity, it also has connections with [8] which are the subject of the first of the four sections. These concern integral equivalence of finitely generated R-modules. where R is an arbitrary noetherian ring. We therefore introduce a finitely generated R-module M and relate to it a short exact sequence (s.e.s.),where F is a free module generated by m elements u1,…, um, and L is generated by elements yj, (j = 1, …, n), of F. We identify the elements u1, …, um with a set of indeterminates X1, …, Xm, and F with the R-module S1 of elements of degree 1 of the graded ring S = R[X1, …, Xm].

On graded special radicals of graded rings

2018 ◽  
Vol 17 (06) ◽  
pp. 1850109 ◽  
Author(s):  
Emil Ilić-Georgijević
Keyword(s):  
Jacobson Radical ◽  
Homogeneous Ideal ◽  
Graded Rings ◽  
Graded Ring ◽  
Direct Sum

In this paper, a graded ring is a ring which is the direct sum of a family of its additive subgroups indexed by a nonempty set under the assumption that the product of homogeneous elements is again homogeneous. We study graded special radicals and special radicals of graded rings, but which contain the corresponding Jacobson radicals. There are two versions of this graded radical, which we name the graded over-Jacobson and the large graded over-Jacobson radical. We establish several characterizations of the graded over-Jacobson radical of a graded ring and also prove that the largest homogeneous ideal contained in the corresponding classical radical of a graded ring coincides with the large graded over-Jacobson radical of that ring.

scholarly journals On graded Brown–McCoy radicals of graded rings

2016 ◽  
Vol 15 (08) ◽  
pp. 1650143 ◽  
Author(s):  
Emil Ilić-Georgijević
Keyword(s):  
Homogeneous Ideal ◽  
Graded Rings ◽  
Graded Ring ◽  
Direct Sum

We investigate the graded Brown–McCoy and the classical Brown–McCoy radical of a graded ring, which is the direct sum of a family of its additive subgroups indexed by a nonempty set, under the assumption that the product of homogeneous elements is again homogeneous. There are two kinds of the graded Brown–McCoy radical, the graded Brown–McCoy and the large graded Brown–McCoy radical of a graded ring. Several characterizations of the graded Brown–McCoy radical are given, and it is proved that the large graded Brown–McCoy radical of a graded ring is the largest homogeneous ideal contained in the classical Brown–McCoy radical of that ring.

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).

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

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.

Primary decomposition of homogeneous ideal in idealization of a module

2018 ◽  
Vol 55 (3) ◽  
pp. 345-352
Author(s):  
Tran Nguyen An
Keyword(s):  
Noetherian Ring ◽  
Primary Decomposition ◽  
Associated Primes ◽  
Homogeneous Ideal ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Irreducible Decomposition

Let R be a commutative Noetherian ring, M a finitely generated R-module, I an ideal of R and N a submodule of M such that IM ⫅ N. In this paper, the primary decomposition and irreducible decomposition of ideal I × N in the idealization of module R ⋉ M are given. From theses we get the formula for associated primes of R ⋉ M and the index of irreducibility of 0R ⋉ M.

Some Remarks on Noetherian Rings

1957 ◽  
Vol 9 ◽  
pp. 35-37
Author(s):  
Michio Yoshida
Keyword(s):  
Present Note ◽  
Noetherian Rings ◽  
Important Theorem ◽  
The University ◽  
Special Case

In his lecture at the University of Kyoto on September 23, 1955, Professor Artin gave an important theorem on Noetherian rings, which seems to have not a few interesting consequences. It is the purpose of our present note to point out one of them. We begin by quoting a special case of the theorem.

Strict Mittag–Leffler modules over Gorenstein injective modules

2019 ◽  
Vol 19 (03) ◽  
pp. 2050050 ◽  
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan
Keyword(s):  
Noetherian Ring ◽  
Direct Limit ◽  
Injective Module ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Injective Modules ◽  
Pure Submodules ◽  
Covering Class ◽  
Gorenstein Injective ◽  
Finitely Presented

In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class [Formula: see text] of Gorenstein injective modules. To this aim, we introduce the notion of [Formula: see text]-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented [Formula: see text]-projective modules, then it is a strict Mittag–Leffler module over [Formula: see text]. As applications, we prove that if [Formula: see text] is a two-sided noetherian ring, then [Formula: see text] is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over [Formula: see text].

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