Let $\mathbb{K}$ be a finite commutative ring, and let $\mathbb{L}$ be a commutative $\mathbb{K}$-algebra. Let $A$ and $B$ be two $n \times n$-matrices over $\mathbb{L}$ that have the same characteristic polynomial. The main result of this paper states that the set $\left\{ A^0,A^1,A^2,\ldots\right\}$ is finite if and only if the set $\left\{ B^0,B^1,B^2,\ldots\right\}$ is finite. We apply this result to Cellular Automata (CA). Indeed, it gives a complete and easy-to-check characterization of sensitivity to initial conditions and equicontinuity for linear CA over the alphabet $\mathbb{K}^n$ for $\mathbb{K} = \mathbb{Z}/m\mathbb{Z}$ i.e., CA in which the local rule is defined by $n\times n$-matrices with elements in $\mathbb{Z}/m\mathbb{Z}$. To prove our main result, we derive an integrality criterion for matrices that is likely of independent interest. Namely, let $\mathbb{K}$ be any commutative ring (not necessarily finite), and let $\mathbb{L}$ be a commutative $\mathbb{K}$-algebra. Consider any $n \times n$-matrix $A$ over $\mathbb{L}$. Then, $A \in \mathbb{L}^{n \times n}$ is integral over $\mathbb{K}$ (that is, there exists a monic polynomial $f \in \mathbb{K}\left[t\right]$ satisfying $f\left(A\right) = 0$) if and only if all coefficients of the characteristic polynomial of $A$ are integral over $\mathbb{K}$. The proof of this fact relies on a strategic use of exterior powers (a trick pioneered by Gert Almkvist). Furthermore, we extend the decidability result concerning sensitivity and equicontinuity to the wider class of additive CA over a finite abelian group. For such CA, we also prove the decidability of injectivity, surjectivity, topological transitivity and all the properties (as, for instance, ergodicity) that are equivalent to the latter.

中文翻译：

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矩阵的完整性、矩阵半群的有限性以及线性和加性元胞自动机的动力学

令 $\mathbb{K}$ 是一个有限交换环，令 $\mathbb{L}$ 是一个交换 $\mathbb{K}$-代数。设 $A$ 和 $B$ 是 $\mathbb{L}$ 上具有相同特征多项式的两个 $n \times n$-矩阵。本文的主要结果表明集合 $\left\{ A^0,A^1,A^2,\ldots\right\}$ 是有限的当且仅当集合 $\left\{ B^0 ,B^1,B^2,\ldots\right\}$ 是有限的。我们将此结果应用于元胞自动机 (CA)。事实上，它提供了一个完整且易于检查的特征，描述了字母表上线性 CA 对初始条件和等连续性的敏感性 $\mathbb{K}^n$ for $\mathbb{K} = \mathbb{Z}/m \mathbb{Z}$ 即，CA 其中局部规则由 $n\times n$-矩阵定义，其中元素在 $\mathbb{Z}/m\mathbb{Z}$ 中。为了证明我们的主要结果，我们为可能具有独立兴趣的矩阵推导出完整性标准。即，让 $\mathbb{K}$ 是任何交换环（不一定是有限的），并让 $\mathbb{L}$ 是一个交换 $\mathbb{K}$-代数。考虑在 $\mathbb{L}$ 上的任何 $n \times n$-matrix $A$。那么，$A \in \mathbb{L}^{n \times n}$ 在 $\mathbb{K}$ 上是积分的（即存在一个单数多项式 $f \in \mathbb{K}\left[ t\right]$ 满足 $f\left(A\right) = 0$) 当且仅当 $A$ 的特征多项式的所有系数在 $\mathbb{K}$ 上都是整数。这一事实的证明依赖于对外部力量的战略性使用（由 Gert Almkvist 首创的技巧）。此外，我们将关于敏感性和等连续性的可判定性结果扩展到有限阿贝尔群上更广泛的加性 CA 类。对于这样的 CA，