Rings with a bounded graded index of nilpotency

This article is devoted to exploration of graded rings with a bounded graded index of nilpotency. It is shown that the graded case is drastically different from the non-graded one, except for gr-semiprime rings. Graded ring is an algebraical object which is a generalization of such structures as polynomial rings and group algebras, and despite being quite obvious and straightforward, graded rings started being explored only in the middle of the 20th century. Construction of this object is based on two simple ideas: the first one is that any element of a ring is a sum of homogenous components, and the second one is that multiplication of homogenous elements induces a group (or a semigroup) structure on homogenous subgroups of the additive group of a ring. In the theory of graded rings, a lot of graded analogues of classic concepts are introduced. For example, an ideal is called a graded ideal if it includes, with any of its elements, its homogenous components; a ring is called a gr-division ring (or a gr-field in the commutative case) if every its nonzero homogenous element is invertible, etc. In this article we consider gr-prime rings (rings without graded nonzero ideals - divisors of zero), gr-semiprime rings (rings without graded nonzero nilpotent ideals), gr-reduced rings (rings without nonzero homogenous nilpotent elements) and, certainly, rings with bounded graded index of nilpotency (rings without nonzero homogenous nilpotent elements with nilpotency degree more than a certain natural n).

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