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# On the Existence of the Graded Exponent for Finite Dimensional $\mathbb{Z}_p$-graded Algebras

Published:2011-05-30
Printed: Jun 2012
• M. Onofrio Di Vincenzo,
Dipartimento di Matematica e Informatica, Università degli Studi della Basilicata, Viale dell'Ateneo Lucano 10, 85100 Potenza, Italia
• Vincenzo Nardozza,
Dipartimento di Matematica, Università degli Studi di Bari, via Orabona 4, 70125 Bari, Italia
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## Abstract

Let $F$ be an algebraically closed field of characteristic zero, and let $A$ be an associative unitary $F$-algebra graded by a group of prime order. We prove that if $A$ is finite dimensional then the graded exponent of $A$ exists and is an integer.
 Keywords: exponent, polynomial identities, graded algebras
 MSC Classifications: 16R50 - Other kinds of identities (generalized polynomial, rational, involution) 16R10 - $T$-ideals, identities, varieties of rings and algebras 16W50 - Graded rings and modules

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