quasiregularity
An element $x$ of a ring is called right quasiregular [resp. left quasiregular] if there is an element $y$ in the ring such that $x+y+xy=0$ [resp. $x+y+yx=0$].
For calculations with quasiregularity, it is useful to introduce the operation^{} $*$ defined:
$$x*y=x+y+xy.$$ 
Thus $x$ is right quasiregular 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 quasiregular if it is both left and right quasiregular. 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 quasiinverse of $x$.
An ideal (one or twosided) of a ring is called quasiregular if each of its elements is quasiregular. Similarly, a ring is called quasiregular if each of its elements is quasiregular (such rings cannot have an identity element^{}).
Lemma.
Let $A$ be an ideal (one or twosided) in a ring $R$. If each element of $A$ is right quasiregular, then $A$ is a quasiregular ideal.
This lemma means that there is no extra generality gained in defining terms such as right quasiregular left ideal^{}, etc.
Quasiregularity is important because it provides elementary characterizations^{} of the Jacobson radical^{} for rings without an identity element:

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The Jacobson radical of a ring is the sum of all quasiregular left (or right) ideals.

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The Jacobson radical of a ring is the largest quasiregular ideal of the ring.
For rings with an identity element, note that $x$ is [right, left] quasiregular if and only if $1+x$ is [right, left] invertible in the ring.
Title  quasiregularity 
Canonical name  Quasiregularity 
Date of creation  20130322 13:12:59 
Last modified on  20130322 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  quasiregular 
Defines  right quasiregular 
Defines  left quasiregular 
Defines  quasiinverse 
Defines  quasiregular ideal 
Defines  quasiregular ring 