Matrix type of some algebras over a field of characteristic \(p\)
All algebras considered in the paper are over a field \(F\) of positive characteristic \(p\). The author proved [in Int. J. Algebra Comput. 5, No. 2, 189-197 (1995; Zbl 0835.16019)] that if \(A\) is a PI algebra then it satisfies all multilinear identities of the matrix algebra \(M_m(F)\). Call the least \(m\) with this property the matrix type of \(A\). Note that such theorem does not hold in characteristic 0 (the infinite dimensional Grassmann algebra \(G\) is such an example). Let \(t_n\) be the matrix type of the algebra \(M_n(G)\). The main result of the present paper is that \(t_n=pn\) (Theorem 2). It follows in turn from another theorem that is of interest as well. Let \(d_k\) be the least degree of a standard identity satisfied by \(M_k(G)\). (It should be noted that according to another result of the author [Isr. J. Math. 81, No. 3, 343-355 (1993; Zbl 0795.16017)] every PI algebra satisfies some standard identity; this is a particular case of the above mentioned result.) Namely Theorem 1 states that for \(k>1\) one has the inequality \(d_k>(2k-3)p-1\). An interesting open problem is stated in the paper. Find a multilinear polynomial that is an identity for the matrix algebra \(M_{np-1}(F)\) but is not for \(M_n(G)\). It is mentioned that even for \(n=1\) it is quite difficult to compute such polynomials. (Their existence is guaranteed by Theorem 2 of the paper.) In fact for \(n=1\) and \(p=3\) such polynomial was obtained in a paper by \textit{A. Giambruno} and the reviewer [Isr. J. Math. 122, 305-316 (2001; Zbl 0990.16022)].
scientific article; zbMATH DE number 1846025