Quasiregular element

In mathematics, specifically ring theory, the notion of quasiregularity provides a computationally convenient way to work with the Jacobson radical of a ring. Intuitively, quasiregularity captures what it means for an element of a ring to be "bad"; that is, have undesirable properties. Although a "bad element" is necessarily quasiregular, quasiregular elements need not be "bad", in a rather vague sense. In this article, we primarily concern ourselves with the notion of quasiregularity for unital rings. However, one section is devoted to the theory of quasiregularity in non-unital rings, which constitutes an important aspect of noncommutative ring theory.
Definition.
Let "R" be a ring (with unity) and let "r" be an element of "R". Then "r" is said to be quasiregular, if 1 − "r" is a unit in "R"; that is, invertible under multiplication. The notions of right or left quasiregularity correspond to the situations where 1 − "r" has a right or left inverse, respectively.
An element "x" of a non-unital ring is said to be right quasiregular if there is "y" such that formula_1. The notion of a left quasiregular element is defined in an analogous manner. The element "y" is sometimes referred to as a right quasi-inverse of "x". If the ring is unital, this definition quasiregularity coincides with that given above. If one writes formula_2, then this binary operation formula_3 is associative. In fact, the map formula_4 (where × denotes the multiplication of the ring "R") is a monoid isomorphism. Therefore, if an element possesses both a left and right quasi-inverse, they are equal.
Note that some authors use different definitions. They call an element "x" right quasiregular if there exists "y" such that formula_5, which is equivalent to saying that 1 + "x" has a right inverse when the ring is unital. If we write formula_6, then formula_7, so we can easily go from one set-up to the other by changing signs. For example, "x" is right quasiregular in one set-up iff −"x" is right quasiregular in the other set-up.
Generalization to semirings.
The notion of quasiregular element readily generalizes to semirings. If "a" is an element of a semiring "S", then an affine map from "S" to itself is formula_17. An element "a" of "S" is said to be right quasiregular if formula_18 has a fixed point, which need not be unique. Each such fixed point is called a left quasi-inverse of "a". If "b" is a left quasi-inverse of "a" and additionally "b" = "ab" + 1, then "b" it is called a quasi-inverse of "a"; any element of the semiring that has a quasi-inverse is said to be quasiregular. It is possible that some but not all elements of a semiring be quasiregular; for example, in the semiring of nonegative reals with the usual addition and multiplication of reals, formula_18 has the fixed point formula_20 for all "a" < 1, but has no fixed point for "a" ≥ 1. If every element of a semiring is quasiregular then the semiring is called a quasi-regular semiring, closed semiring, or occasionally a Lehmann semiring (the latter honoring the paper of Daniel J. Lehmann.)
Examples of quasi-regular semirings are provided by the Kleene algebras (prominently among them, the algebra of regular expressions), in which the quasi-inverse is lifted to the role of a unary operation (denoted by "a"*) defined as the least fixedpoint solution. Kleene algebras are additively idempotent but not all quasi-regular semirings are so. We can extend the example of nonegative reals to include infinity and it becomes a quasi-regular semiring with the quasi-inverse of any element "a" ≥ 1 being the infinity. This quasi-regular semiring is not additively idempotent however, so it is not a Kleene algebra. It is however a complete semiring. More generally, all complete semirings are quasiregular. The term "closed semiring" is actually used by some authors to mean complete semiring rather than just quasiregular.
Conway semirings are also quasiregular; the two Conway axioms are actually independent, i.e. there are semirings satisfying only the product-star [Conway] axiom, ("ab")* = 1+"a"("ba")*"b", but not the sum-star axiom, ("a"+"b")* = ("a"*"b")*"a"* and vice versa; it is the product-star [Conway] axiom that implies that a semiring is quasiregular. Additionally, a commutative semiring is quasiregular if and only if it satisfies the product-star Conway axiom.
Quasiregular semirings appear in s, a generalization of the shortest path problem.