一个抽象分解定理及其应用
An Abstract Factorization Theorem and Some Applications
摘要：
我们结合语言的幺半群与语言的前序，以完善一些基本方面的经典理论的因式分解和证明一个抽象的因式分解定理与各种应用。特别地，我们得到了一个关于“原子因子分解”的经典定理的推广，从可消到Dedekind有限（交换或非交换）幺半群，该定理可以追溯到P.M. Cohn在20世纪60年代的工作;恢复了D.D. Anderson和S. Valdes-Leon关于交换环中的“不可约因子分解”;安东尼奥和作者，在某些“集合的幺半群”自然产生的加性数论和算术组合的特点原子性;给出了一个么半群理论证明，（交换或非交换）环$R$是n个不可分解模的直和（这实际上是具有有限乘积的某些范畴的对象的更一般分解定理的特殊情况，其中不可分解的$R$ -模被刻画为合适的“模幺半群”的原子）。
英文摘要：
We combine the language of monoids with the language of preorders so as to refine some fundamental aspects of the classical theory of factorization and prove an abstract factorization theorem with a variety of applications. In particular, we obtain a generalization, from cancellative to Dedekind-finite (commutative or non-commutative) monoids, of a classical theorem on "atomic factorizations" that traces back to the work of P.M. Cohn in the 1960s; recover a theorem of D.D. Anderson and S. Valdes-Leon on "irreducible factorizations" in commutative rings; improve on a theorem of A.A. Antoniou and the author that characterizes atomicity in certain "monoids of sets" naturally arising from additive number theory and arithmetic combinatorics; and give a monoid-theoretic proof that every module of finite uniform dimension over a (commutative or non-commutative) ring $R$ is a direct sum of finitely many indecomposable modules (this is in fact a special case of a more general decomposition theorem for the objects of certain categories with finite products, where the indecomposable $R$-modules are characterized as the atoms of a suitable "monoid of modules").
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