Abstract factorization theorems with applications to idempotent factorizations

Let  be a preorder on a monoid H with identity 1H 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, . . . , xk of -non-units of H with x1 = x, where u ∈ H is a -unit if u  1H  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 1R) and (ii) a  b if and only if the right annihilator of 1R −b is contained in the right annihilator of 1R − 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 1R − 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 RR. 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).