quasi-regularity
An element of a ring is called right quasi-regular [resp. left quasi-regular] if there is an element in the ring such that [resp. ].
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).
Lemma.
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.
Quasi-regularity 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 quasi-regular left (or right) ideals.
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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.
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 |