A Basis for the Graded Identities of the Matrix Algebra of Order Two over a Finite Field of Characteristic p≠2

Let K be a finite field of characteristic p>2, and let M2(K) be the matrix algebra of order two over K. We describe up to a graded isomorphism the 2-gradings of M2(K). It turns out that there are only two nonisomorphic nontrivial such gradings. Furthermore, we exhibit finite bases of the graded polynomial identities for each one of these two gradings. One can distinguish these two gradings by means of the graded polynomial identities they satisfy.