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Graded identities for algebras with elementary gradings over an infinite field
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 1 / February 2022
- Published online by Cambridge University Press:
- 10 January 2022, pp. 146-166
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- Article
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Let $F$
be an infinite field of positive characteristic $p > 2$
and let $G$
be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$
-grading. Let $E$
be the infinite-dimensional Grassmann algebra. For every $a$
, $b\in \mathbb {N}$
, we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$
, as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$
and $M_{ar+bs,as+br}(E)$
are $F$
-algebras which are not PI equivalent. Actually, we prove that the $T_{G}$
-ideal of the former algebra is contained in the $T$
-ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.
