[Paper Review] Decomposing Finite $\mathbb{Z}$-Algebras

Martin Kreuzer, Alexei Miasnikov+1 more|arXiv (Cornell University)|Aug 3, 2023

Advanced Topics in AlgebraMathematics3 citations

TL;DR

This paper presents an algorithmic decomposition of finite $Χ$-algebras by constructing the maximal ring of scalars of their multiplication map, decomposing it into indecomposable components via primitive idempotents, and lifting the decomposition to $R/\operatorname{Ann}(R)$. The method runs in probabilistic polynomial time with one integer factorization, enabling direct decomposition of $R$ when $\operatorname{Ann}(R) = 0$. The key contribution is a fully algorithmic and complexity-aware approach to algebra decomposition for non-associative, non-unital rings with finitely generated additive groups.

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 $R/{ m Ann}(R)$ 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 $R/{ m Ann}(R)$. 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 ${ m Ann}(R) = 0$, these algorithms compute a direct decomposition of $R$ into directly indecomposable factors.

Motivation & Objective

• To develop an explicit, algorithmic method for decomposing finite $Χ$-algebras into directly indecomposable factors.
• To address the challenge of decomposing non-associative, non-unital rings with finitely generated additive groups.
• To track the computational complexity of each step, particularly identifying the need for integer factorization as the main complexity bottleneck.
• To establish conditions under which decompositions of $R/\operatorname{Ann}(R)$ can be lifted to $R$, and when they cannot.
• To provide a framework for lifting bilinear map decompositions to algebraic decompositions using scalar rings and idempotents.

Proposed method

• Construct the maximal ring of scalars $\mathfrak{S}(f)$ for the bilinear multiplication map $f: R/\operatorname{Ann}_\lambda(R) \times R/\operatorname{Ann}_\varrho(R) \to R^2$ via solving linear Diophantine equations.
• Present $\mathfrak{S}(f)$ as a $\mathbb{Z}$-algebra via generators and relations using algorithmic computation of defining equations.
• Compute primitive idempotents of the maximal ring of scalars using a simplified primary decomposition algorithm and spectral decomposition over $\operatorname{Spec}(S)$.
• Decompose the bilinear map $f$ into directly indecomposable components using the primitive idempotents of $\mathfrak{S}(f)$.
• Lift the decomposition of the bilinear map to a direct product decomposition of $R/\operatorname{Ann}(R)$ via module-theoretic lifting techniques.
• Establish criteria under which the decomposition of $R/\operatorname{Ann}(R)$ can be lifted to a decomposition of $R$, and identify obstructions via counterexamples.

Experimental results

Research questions

• RQ1 Can the decomposition of a finite $\mathbb{Z}$-algebra $R$ be computed algorithmically when $R$ is not necessarily associative or unital?
• RQ2 What is the computational complexity of decomposing $R/\operatorname{Ann}(R)$ into directly indecomposable factors, and how does it depend on integer factorization?
• RQ3 Under what conditions can a decomposition of $R/\operatorname{Ann}(R)$ be lifted to a decomposition of $R$?
• RQ4 How can the maximal ring of scalars of a bilinear map be computed algorithmically and used to decompose the map?
• RQ5 Can the decomposition of the Lie ring associated to a nilpotent group of class 2 be used to recover a decomposition of the group itself?

Key findings

• The maximal ring of scalars $\mathfrak{S}(f)$ of a bilinear map $f$ can be computed in polynomial time via solving linear Diophantine equations.
• The decomposition of the maximal ring of scalars into indecomposable components is achieved via computation of primitive idempotents, which is possible in probabilistic polynomial time with one integer factorization.
• The decomposition of the bilinear map $f$ lifts to a direct product decomposition of $R/\operatorname{Ann}(R)$, with the entire procedure running in probabilistic polynomial time plus one integer factorization.
• When $\operatorname{Ann}(R) = 0$, the algorithm computes a direct decomposition of $R$ into directly indecomposable $\mathbb{Z}$-algebras.
• Counterexamples show that decompositions of $R/\operatorname{Ann}(R)$ cannot always be lifted to $R$, particularly when $\operatorname{Ann}(R) \cap R^2$ is not a direct summand.
• For nilpotent groups of class 2, the associated Lie ring $L(G)$ allows decomposition of $G/\operatorname{Z}(G)$ into cyclic subgroups via the bilinear map $f_G$, but lifting to $G$ remains an open question.