Abstract Factorization Theorems with Applications to Idempotent Factorizations
Laura Cossu, Salvatore Tringali
Abstract
Let $\preceq$ be a preorder on a monoid $H$ and $s$ be an integer $\ge 2$. The $\preceq$-height of an $x \in H$ is the sup of the integers $k \ge 1$ for which there is a (strictly) $\preceq$-decreasing sequence $x_1,\ldots,x_k$ of $\preceq$-non-units of $H$ with $x_1 = x$ (with $\sup\emptyset:=0$), where $u\in H$ is a $\preceq$-unit if $u\preceq 1_H\preceq u$ and a $\preceq$-non-unit otherwise. We say $H$ is $\preceq$-artinian if there exists no $\preceq$-decreasing sequence $x_1,x_2,\ldots$ of elements of $H$; and strongly $\preceq$-artinian if the $\preceq$-height of each element is finite. We establish that, if $H$ is $\preceq$-artinian, then each $\preceq$-non-unit $x\in H$ factors through the $\preceq$-irreducibles of degree $s$, where a $\preceq$-irreducible of degree $s$ is a $\preceq$-non-unit $a\in H$ that cannot be written as a product of $s$ or fewer $\preceq$-non-units each of which is (strictly) smaller than $a$ with respect to $\preceq$. In addition, we show that, if $H$ is strongly $\preceq$-artinian, then $x$ factors through the $\preceq$-quarks of $H$, where a $\preceq$-quark is a $\preceq$-min $\preceq$-non-unit. In the process, we also obtain upper bounds for the length of a shortest factorization of $x$ (into either $\preceq$-irreducible of degree $s$ or $\preceq$-quarks) in terms of its $\preceq$-height. Next, we specialize these abstract results to the case in which $H$ is the multiplicative submonoid of a ring $R$ formed by the zero divisors and the identity $1_R$, and $\preceq$ is the preorder on $H$ defined by $a\preceq b$ iff $r_R(1_R-b)\subseteq r_R(1_R-a)$, where $r_R(\cdot)$ denotes a right annihilator. We can thus recover and improve on classical theorems of J.A. Erdos (1967), R.J.H. Dawlings (1981), and J. Fountain (1991) on idempotent factorizations in the endomorphism ring of a free module of finite rank over a skew field or a commutative DVD.