Abstract factorization theorems with applications to idempotent factorizations

Let ⪯ be a preorder on a monoid H with identity 1_H and s be an integer ≥ 2. The ⪯-height of an element x ∈ H is the supremum of the integers k ≥ 1 for which there is a (strictly) ⪯-decreasing sequence x₁, …, x_k of ⪯-non-units of H with x₁ = x, where u ∈ H is a ⪯-unit if u ⪯ 1_H ⪯ u and a ⪯-non-unit otherwise. We say H is ⪯-artinian if there is no infinite ⪯-decreasing sequence of elements of H, and strongly ⪯-artinian if the ⪯-height of each element is finite.

We establish that, if H is ⪯-artinian, then each ⪯-non-unit x ∈ H factors through the ⪯-irreducibles of degree s, where a ⪯-irreducible of degree s is a ⪯-non-unit a ∈ H that cannot be written as a product of s or fewer ⪯-non-units each of which is (strictly) smaller than a with respect to ⪯. In addition, we show that, if H is strongly ⪯-artinian, then x factors through the ⪯-quarks of H, where a ⪯-quark is a ⪯-minimal ⪯-non-unit. In the process, we obtain upper bounds for the length of a shortest factorization of x into ⪯-irreducibles of degree s (resp., ⪯-quarks) in terms of its ⪯-height.

Next, we specialize these results to the case in which (i) H is the multiplicative submonoid of a ring R formed by the zero divisors of R (and the identity 1_R) and (ii) a ⪯ b if and only if the right annihilator of 1_R − b is contained in the right annihilator of 1_R − a. If H is ⪯-artinian (resp., strongly ⪯-artinian), then every zero divisor of R factors as a product of ⪯-irreducibles of degree s (resp., ⪯-quarks); and we prove that, for a variety of right Rickart rings, either the ⪯-quarks or the ⪯-irreducibles of degree 2 or 3 are coprimitive idempotents (an idempotent e ∈ R is coprimitive if 1_R − e is primitive). In the latter case, we also derive sharp upper bounds for the length of a shortest idempotent factorization of a zero divisor x ∈ R in terms of the ⪯-height of x and the uniform dimension of R_R. In particular, 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 (e.g., we find that every singular n-by-n matrix over a commutative DVD, with n ≥ 2, is a product of 2n − 2 or fewer idempotent matrices of rank n − 1).