TY - JOUR A1 - Kreuzer, Martin A1 - Miasnikov, Alexei A1 - Walsh, Florian T1 - Decomposing finite Z-algebras JF - Journal of Algebra N2 - 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 scalarsS, and then lifting the decomposition ofSto 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. KW - Algebra decomposition KW - Directly indecomposable factor KW - Bilinear map KW - Maximal ring of scalars KW - Primitive idempotent KW - Lie ring Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus4-19184 SN - 1090-266X VL - 2025 IS - 664 B SP - 206 EP - 246 PB - Elsevier CY - Amsterdam ER -