Maximal Mathieu-Zhao Subspaces of M 2 ( F ) and M n ( F )

Let Formula: see text be a field. In 2010, the second author introduced the definition of a Mathieu subspace (or Mathieu-Zhao subspace) for associative Formula: see text-algebras, but they are not well-understood even for finite-dimensional algebras. In Section 2 of this note we give a complete classification of Mathieu subspaces of Formula: see text for arbitrary fields Formula: see text. In Section 3 we give a construction for a family of maximal Mathieu subspaces of the matrix algebras Formula: see text Formula: see text of codimension Formula: see text for Formula: see text when Formula: see text has characteristic zero or when char. Formula: see text.