For calculations with quasi-regularity, it is useful to introduce the operation defined:
Thus is right quasi-regular if there is an element such that . The operation is easily demonstrated to be associative, and for all .
An element is called quasi-regular if it is both left and right quasi-regular. In this case, there are elements and such that (equivalently, ). A calculation shows that
So is a unique element, depending on , called the quasi-inverse of .
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).
Let be an ideal (one- or two-sided) in a ring . If each element of is right quasi-regular, then 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.
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 is [right, left] quasi-regular if and only if is [right, left] invertible in the ring.
|Date of creation||2013-03-22 13:12:59|
|Last modified on||2013-03-22 13:12:59|
|Last modified by||mclase (549)|