An element x of a ring is called right quasi-regular [resp. left quasi-regular] if there is an element y in the ring such that x+y+xy=0 [resp. x+y+yx=0].

For calculations with quasi-regularity, it is useful to introduce the operationMathworldPlanetmath * defined:


Thus x is right quasi-regular if there is an element y such that x*y=0. The operation * is easily demonstrated to be associative, and x*0=0*x=x for all x.

An element x is called quasi-regular if it is both left and right quasi-regular. In this case, there are elements y and z such that x+y+xy=0=x+z+zx (equivalently, x*y=z*x=0). A calculation shows that


So y=z is a unique element, depending on x, called the quasi-inverse of x.

An ideal (one- or two-sided) of a ring is called quasi-regular if each of its elements is quasi-regular. Similarly, a ring is called quasi-regular if each of its elements is quasi-regular (such rings cannot have an identity elementMathworldPlanetmath).


Let A be an ideal (one- or two-sided) in a ring R. If each element of A is right quasi-regular, then A is a quasi-regular ideal.

This lemma means that there is no extra generality gained in defining terms such as right quasi-regular left idealMathworldPlanetmathPlanetmath, etc.

Quasi-regularity is important because it provides elementary characterizationsMathworldPlanetmath of the Jacobson radicalMathworldPlanetmath for rings without an identity element:

  • The Jacobson radical of a ring is the sum of all quasi-regular left (or right) ideals.

  • The Jacobson radical of a ring is the largest quasi-regular ideal of the ring.

For rings with an identity element, note that x is [right, left] quasi-regular if and only if 1+x is [right, left] invertible in the ring.

Title quasi-regularity
Canonical name Quasiregularity
Date of creation 2013-03-22 13:12:59
Last modified on 2013-03-22 13:12:59
Owner mclase (549)
Last modified by mclase (549)
Numerical id 8
Author mclase (549)
Entry type Definition
Classification msc 16N20
Synonym quasi regular
Synonym quasi regularity
Related topic JacobsonRadical
Related topic RegularIdeal
Related topic HomotopesAndIsotopesOfAlgebras
Defines quasi-regular
Defines right quasi-regular
Defines left quasi-regular
Defines quasi-inverse
Defines quasi-regular ideal
Defines quasi-regular ring