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:

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

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 1   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:

• 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.