Decomposing finite Z-algebras

For a finite Z-algebra R, i.e., for a ring which is not necessarily associative or unitary, but whose additive group is finitely generated, we construct a decomposition of R/Ann(R) into directly indecomposable factors under weak hypotheses. The method is based on constructing and decomposing a ring of scalars S, and then lifting the decomposition of S to the bilinear map given by the multiplication of R, and finally to R/Ann(R). All steps of the construction are given as explicit algorithms and it is shown that the entire procedure has a probabilistic polynomial time complexity in the bit size of the input, except for the possible need to calculate the prime factorization of an integer. In particular, in the case when Ann(R)=0, these algorithms compute direct decompositions of R into directly indecomposable factors.