scholarly journals Morita duality and finitely group-graded rings

1995 ◽  
Vol 52 (2) ◽  
pp. 189-194 ◽  
Author(s):  
Shenggui Zhang
Keyword(s):  
Graded Rings ◽  
Graded Ring ◽  
Matrix Rings

We give the relation between the (rigid) graded Morita duality and the Morita duality on a finitely group-graded ring and the relation between a left Morita ring and some of its matrix rings.

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 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.

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 Complementation of the Jacobson group in a matrix ring

2005 ◽  
Vol 72 (2) ◽  
pp. 317-324
Author(s):  
David Dolžan
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Matrix Ring ◽  
Graded Rings ◽  
Matrix Rings ◽  
Generators And Relations ◽  
Group Of Units ◽  
The Matrix ◽  
Finite Commutative Ring

The Jacobson group of a ring R (denoted by  = (R)) is the normal subgroup of the group of units of R (denoted by G(R)) obtained by adding 1 to the Jacobson radical of R (J(R)). Coleman and Easdown in 2000 showed that the Jacobson group is complemented in the group of units of any finite commutative ring and also in the group of units a n × n matrix ring over integers modulo ps, when n = 2 and p = 2, 3, but it is not complemented when p ≥ 5. In 2004 Wilcox showed that the answer is positive also for n = 3 and p = 2, and negative in all the remaining cases. In this paper we offer a different proof for Wilcox's results and also generalise the results to a matrix ring over an arbitrary finite commutative ring. We show this by studying the generators and relations that define a matrix ring over a field. We then proceed to examine the complementation of the Jacobson group in the matrix rings over graded rings and prove that complementation depends only on the 0-th grade.

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.

scholarly journals Control subgroups and birational extensions of graded rings

1999 ◽  
Vol 22 (2) ◽  
pp. 411-415
Author(s):  
Salah El Din S. Hussein
Keyword(s):  
Prime Ideal ◽  
Zariski Topology ◽  
Graded Rings ◽  
Graded Ring ◽  
Open Set

In this paper, we establish the relation between the concept of control subgroups and the class of graded birational algebras. Actually, we prove that ifR=⊕σ∈GRσis a stronglyG-graded ring andH⊲G, then the embeddingi:R(H)↪R, whereR(H)=⊕σ∈HRσ, is a Zariski extension if and only ifHcontrols the filterℒ(R−P)for every prime idealPin an open set of the Zariski topology onR. This enables us to relate certain ideals ofRandR(H)up to radical.

Cayley graphs of graded rings

2018 ◽  
Vol 17 (06) ◽  
pp. 1850116
Author(s):  
Saadoun Mahmoudi ◽  
Shahram Mehry ◽  
Reza Safakish
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Graded Rings ◽  
Graded Ring ◽  
Total Graph ◽  
Hamming Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Cayley Sum Graph

Let [Formula: see text] be a subset of a commutative graded ring [Formula: see text]. The Cayley graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The Cayley sum graph [Formula: see text] is a graph whose vertex set is [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be the set of homogeneous elements and [Formula: see text] be the set of zero-divisors of [Formula: see text]. In this paper, we study [Formula: see text] (total graph) and [Formula: see text]. In particular, if [Formula: see text] is an Artinian graded ring, we show that [Formula: see text] is isomorphic to a Hamming graph and conversely any Hamming graph is isomorphic to a subgraph of [Formula: see text] for some finite graded ring [Formula: see text].

scholarly journals On graded Ω-groups

Filomat ◽  
2015 ◽  
Vol 29 (10) ◽  
pp. 2167-2183 ◽  
Author(s):  
Emil Ilic-Georgijevic
Keyword(s):  
Prime Radical ◽  
Graded Rings ◽  
Graded Ring ◽  
Partial Structures ◽  
Natural Way

In this paper we study the notion of a graded ?-group (X;+ ?), but graded in the sense of M. Krasner, i.e., we impose nothing on the grading set except that it is nonempty, since operations of and the grading of (X,+) induce operations (generally partial) on the grading set. We prove that graded ?-groups in Krasner?s sense are determined up to isomorphism by their homogeneous parts, which, with respect to induced operations, represent partial structures called ?-homogroupoids, thus narrowing down the theory of graded -groups to the theory of ?-homogroupoids. This approach already proved to be useful in questions regarding A. V. Kelarev?s S-graded rings inducing S; where S is a partial cancellative groupoid. Particularly, in this paper we prove that the homogeneous subring of a Jacobson S-graded ring inducing S is Jacobson under certain assumptions. We also discuss the theory of prime radicals for ?-homogroupoids thus extending results of A. V. Mikhalev, I. N. Balaba and S. A. Pikhtilkov in a natural way. We study some classes of ?-homogroupoids for which the lower and upper weakly solvable radicals coincide and also, study the question of the homogeneity of the prime radical of a graded ring.

Comparing Graded Versions of the Prime Radical

1991 ◽  
Vol 34 (2) ◽  
pp. 158-164 ◽  
Author(s):  
M. A. Beattie ◽  
Liu S.X. ◽  
P. N. Stewart
Keyword(s):  
Prime Radical ◽  
Graded Rings ◽  
Graded Ring ◽  
Associative Rings

AbstractLet G be a group with identity e, let λ be a normal supernilpotent radical in the category of associative rings and let λref be the reflected radical in the category of G-graded rings. Then for A a G-graded ring, λref(A) is the largest graded ideal of A whose intersection with Ae is λ (Ae). For λ = B, the prime radical, we compare Bref(A) to BG(A) = B(A)G, the largest graded ideal in B(A).

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