Radicals of Graded Rings

Jespers, Eric

doi:10.1016/b978-0-444-81528-6.50011-8

Cited by 6 publications

(2 citation statements)

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“…An ideal / of R = 0 R g is said to be homogeneous if / = 0 IC\R g . This problem has not been solved geG gsG even for u.p.-groups (see [7], [8], [10]). Polynomial identities give what appears to be the first natural sufficient condition which is applicable to the case of arbitrary groups.…”

Section: Geg R G R H = R Gh ) For Al\gh E Gmentioning

confidence: 99%

On group graded rings satisfying polynomial identities

Kelarev

Okniński

1995

Glasgow Math. J.

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A number of classical theorems of ring theory deal with nilness and nilpotency of the Jacobson radical of various ring constructions (see [10], [18] This paper is devoted to the radicals of group graded rings, which have been actively investigated by many authors (see [10], [14]). Let G be a group. An associative ring R = 0 ^g is said to be G-graded {stronglyFirst, we consider algebras over a field of characteristic zero. In this case our result will be also of interest in connection with the well-known problem of finding necessary and sufficient conditions for the Jacobson radical to be homogeneous. An ideal / of R = 0 R g is said to be homogeneous if / = 0 IC\R g . This problem has not been solved Pi-algebra over a field of characteristic zero. If the Jacobson radical J(R e ) is nil, then J(R) is a homogeneous nil ideal of R.The following corollary to the main theorem is worth mentioning.COROLLARY 2. Let G be a group with identity e, and let R = 0 R g be a strongly g &G G-graded Pi-algebra over a field of characteristic zero. IfJ(R e ) is nilpotent, then J(R) is nilpotent.It is impossible to replace strongly graded algebras by ordinary group graded algebras in Corollary 2. Indeed, let A be the free commutative algebra with free generators a u a 2 , Denote by / the ideal of A generated by a u a\, a\, Then All is positively graded, and so All = 0 A z where Z is the infinite cyclic group. Although A o = 0, it is zeZ clear that J(A/I) = A/1 is not nilpotent.One cannot omit the restriction on the characteristic of the field neither in Theorem Glasgow Math.

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Section: Geg R G R H = R Gh ) For Al\gh E Gmentioning

confidence: 99%

On group graded rings satisfying polynomial identities

Kelarev

Okniński

1995

Glasgow Math. J.

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“…A ring R is said to be S-graded if R = © s 6 s R s is a direct sum of additive subgroups R s and R s R t c R st for all s, t e S. If S is a semigroup with zero 0, and R = © s 6 s R s is an S-graded ring such that R o = 0, then R is said to be a contracted S-graded ring (see [6,7]). Every S-graded ring can be viewed as a contracted S°-graded ring, where S° is obtained by adjoining a zero 0 to S. Semigroupgraded rings include as special cases many other ring constructions, including group-graded rings and Morita contexts (see [14]). Denote by B n the semigroup consisting of zero and all the standard nxn matrix units.…”

Section: Introductionmentioning

confidence: 99%