Algebraic structures connected with pairs of compatible associative algebras

We study associative multiplications in semisimple associative algebras over compatible with the usual one or, in other words,linear deformations of semi-simple associative algebras over . It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures,which we call M-structures in the matrix case and PM -structures in the case of direct sums of several matrix algebras. We also investigate various properties of PM -structures, provide numerous examples and describe an important class of PM -structures. The classification of these PM -structures naturally leads to affine Dynkin diagrams of A,D,E-types.