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arXiv · 2308.01735

Decomposing Finite $\mathbb{Z}$-Algebras

Martin Kreuzer·Alexei Miasnikov·Florian Walsh
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Abstract

For a finite $\mathbb{Z}$-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 $R/{\rm Ann}(R)$ 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 $R/{\rm Ann}(R)$. 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 one integer. In particular, in the case when ${\rm Ann}(R) = 0$, these algorithms compute a direct decomposition of $R$ into directly indecomposable factors.

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Martin Kreuzer, Alexei Miasnikov, Florian Walsh. 2023-08-03. Decomposing Finite $\mathbb{Z}$-Algebras. https://arxiv.org/abs/2308.01735

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