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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 1 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.
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