Geodesic
Gradings and Graded Identities for the Matrix Algebra of Order Two in Characteristic 2
Serdica Mathematical Journal, Tome 38 (2012) no. 1-3, pp. 189-198.

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Let K be an infinite field and let M2(K) be the matrix algebra of order two over K. The polynomial identities of M2(K) are known whenever the characteristic of K is different from 2. The algebra M2(K) admits a natural grading by the cyclic group of order 2; the graded identities for this grading are known as well. But M2(K) admits other gradings that depend on the field and on its characteristic. Here we describe the graded identities for all nontrivial gradings by the cyclic group of order 2 when the characteristic of K equals 2. It turns out that there is only one grading to consider. This grading is not elementary. On the other hand the graded identities are the same as for the elementary grading.
Keywords: Gradings on Matrix Algebras, Graded Identities, Polynomial Identities in characteristic two 
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Koshlukov, Plamen; César dos Reis, Júlio. Gradings and Graded Identities for the Matrix Algebra of Order Two in Characteristic 2. Serdica Mathematical Journal, Tome 38 (2012) no. 1-3, pp. 189-198. http://geodesic.mathdoc.fr/item/SMJ2_2012_38_1-3_a8/