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Abstract Factorization Theorems with Applications to Idempotent Factorizations

Laura Cossu, Salvatore Tringali

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.

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