Semigroup Graded Rings and Jacobson Rings

A NOTE ON NIL AND JACOBSON RADICALS IN GRADED RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813501211 ◽

2014 ◽

Vol 13 (04) ◽

pp. 1350121 ◽

Cited By ~ 12

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.

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On graded rings and modules of quotients

Communications in Algebra ◽

10.1080/00927877808822328 ◽

1978 ◽

Vol 6 (18) ◽

pp. 1923-1959 ◽

Cited By ~ 16

Author(s):

Van F. Oystaeyen

Keyword(s):

Graded Rings

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Generators of graded rings of modular forms

Journal of Number Theory ◽

10.1016/j.jnt.2013.12.008 ◽

2014 ◽

Vol 138 ◽

pp. 97-118 ◽

Cited By ~ 4

Author(s):

Nadim Rustom

Keyword(s):

Modular Forms ◽

Graded Rings

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

Algebra and Logic ◽

10.1007/s10469-018-9507-4 ◽

2018 ◽

Vol 57 (5) ◽

pp. 353-359

Author(s):

A. L. Kanunnikov

Keyword(s):

Graded Rings

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A Note on Local Reduction Numbers and $a^*$-Invariants of Graded Rings

Tokyo Journal of Mathematics ◽

10.3836/tjm/1244208478 ◽

2004 ◽

Vol 27 (1) ◽

pp. 113-124

Author(s):

Qu\^oc Vi\d{\^e}t Duong

Keyword(s):

Graded Rings ◽

Local Reduction

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Filtered and Graded Rings

Lectures on Algebraic Quantum Groups ◽

10.1007/978-3-0348-8205-7_12 ◽

2002 ◽

pp. 103-111

Author(s):

Ken A. Brown ◽

Ken R. Goodearl

Keyword(s):

Graded Rings

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On Strongly Groupoid Graded Rings and the Corresponding Clifford Theorem

Algebra Colloquium ◽

10.1142/s1005386706000198 ◽

2006 ◽

Vol 13 (02) ◽

pp. 181-196 ◽

Cited By ~ 4

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.

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