quasi-regularity

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 operation $*$ defined:

$$x*y=x+y+xy.$$

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

$$y=0*y=(z*x)*y=z*(x*y)=z.$$

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 element).

Lemma.

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 ideal, etc.

Quasi-regularity is important because it provides elementary characterizations of the Jacobson radical 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