  We study associative multiplications in semi-simple associative algebras over
C compatible with the usual one or, in other words, linear deformations of
semi-simple associative algebras over C. 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-type.
