A non-zero \(R\)-module \(M\) is called compressible if it can be embedded into each of its non-zero submodules. A compressible module is called critically compressible if it cannot be embedded in any of its proper factor modules. Call an algebra weakly primitive if it is a weakly primitive ring, that is, if it has a faithful critically compressible module. The author proves that the Kronecker product of two weakly primitive algebras \(R_ 1 \otimes_ F R_ 2\) over a field \(F\) is a weakly primitive algebra if and only if \(D_ 1 \otimes_ F D_ 2\) is a weakly primitive algebra, where \(D_ i\) \((i = 1,2,)\) are the centralizers of \(R_ 1\) and \(R_ 2\) respectively. Some results of Azumaya and Nakayama on primitive algebras are also generalized to weakly primitive algebras by introducing the notion of weak socles.