Graded Central Polynomials for T-Prime Algebras

Abstract

Let K be a field, char K = 0, and let E = E ₀⊕ E ₁ be the Grassmann algebra of infinite dimension over K, equipped with its natural ℤ₂-grading. If G is a finite abelian group and R = ⨁ _g∈G R ^(g) is a G-graded K-algebra, then the algebra R⊗ E can be G × ℤ₂-graded by setting (R⊗ E)^{(g, i)} = R ^(g) ⊗ E _i . In this article we describe the graded central polynomials for the T-prime algebras M _n (E)≅ M _n (K)⊗ E. As a corollary we obtain the graded central polynomials for the algebras M _{a, b} (E)⊗ E. As an application, we determine the ℤ₂-graded identities and central polynomials for E⊗ E.

Key Words:

• Central polynomials
• Graded central polynomials
• Graded identities
• Graded T-ideal
• Graded T-space

2000 Mathematics Subject Classification:

• 16R10
• 16R50
• 16R99

ACKNOWLEDGMENT

The authors are thankful to the referee, whose comments improved much of the exposition.

The first author was partially supported by CNPq Grant 620025/2006-9, the second author by CNPq Grant 620025/2006-9, and the third author by CNPq Grant 302655/2005-0, and by FAPESP Grants 04/13766-2 and 05/60337-2.

Notes

Communicated by A. Elduque.