Graded central polynomials for the matrix algebra of order two

摘要

Let K be an infinite integral domain, and let A = M ₂(K) be the matrix algebra of order two over K. The algebra A can be given a natural -grading by assuming that the diagonal matrices are the 0-component while the off-diagonal ones form the 1-component. In this paper we study the graded identities and the graded central polynomials of A. We exhibit finite bases for these graded identities and central polynomials. It turns out that the behavior of the graded identities and central polynomials in the case under consideration is much like that in the case when K is an infinite field of characteristic 0 or p > 2. Our proofs are characteristic-free so they work when K is an infinite field, char K = 2. Thus we describe finite bases of the graded identities and graded central polynomials for M ₂(K) in this case as well. Keywords Graded identities - Graded central polynomials - Graded T-ideal - Graded T-space - Basis of identities Mathematics Subject Classification (2000) 16R10 - 16R50 - 16R99 Communicated by D. Segal.