In mathematics, specifically ring theory, the notion of quasiregularity provides a computationally convenient way to work with the Jacobson radical of a ring.
[1] 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.
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.
[1] An element x of a non-unital ring R is said to be right quasiregular if there exists y in R such that
[2] The notion of a left quasiregular element is defined in an analogous manner.
[3] If the ring is unital, this definition of quasiregularity coincides with that given above.
(where × denotes the multiplication of the ring R) is a monoid isomorphism.
[4] Therefore, if an element possesses both a left and right quasi-inverse, they are equal.
[8] The notion of quasiregular element readily generalizes to semirings.
Each such fixed point is called a left quasi-inverse of a.
[19]) 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.
This quasi-regular semiring is not additively idempotent however, so it is not a Kleene algebra.
Additionally, a commutative semiring is quasiregular if and only if it satisfies the product-star Conway axiom.