Graphs over Graded Rings and Relation with Hamming Graph

2021 ◽  
Author(s):  
Shahram Mehry ◽  
Saadoun Mahmoudi
Keyword(s):  
Graded Rings ◽  
Hamming Graph

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 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 graded rings and modules of quotients

1978 ◽  
Vol 6 (18) ◽  
pp. 1923-1959 ◽  
Author(s):  
Van F. Oystaeyen
Keyword(s):  
Graded Rings

scholarly journals Generators of graded rings of modular forms

2014 ◽  
Vol 138 ◽  
pp. 97-118 ◽  
Author(s):  
Nadim Rustom
Keyword(s):  
Modular Forms ◽  
Graded Rings

Criteria for the Validity of Goldie’s Theorems for Graded Rings

2018 ◽  
Vol 57 (5) ◽  
pp. 353-359
Author(s):  
A. L. Kanunnikov
Keyword(s):  
Graded Rings

Filtered and Graded Rings

2002 ◽  
pp. 103-111
Author(s):  
Ken A. Brown ◽  
Ken R. Goodearl
Keyword(s):  
Graded 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 Finite domination and Novikov homology over strongly ℤ-graded rings

2017 ◽  
Vol 221 (2) ◽  
pp. 661-685 ◽  
Author(s):  
Thomas Hüttemann ◽  
Luke Steers
Keyword(s):  
Graded Rings
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