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In this paper we show that part of the structure of an artinian ring is determined when it has a solvable, simple, nilpotent, supersolvable, torsion, or finitely generated quasi-regular group. For the case of a simple quasi-regular group, the rings are completely described. We also give an application of our results to finite dimensional group algebras.Our proofs depend, in part, on properties of matrix algebras over division rings and the following two known lemmas. Lemma 1. Let R be an associative ring and let n be a fixed natural number such that (i) R = Mx®M2® ■ ■ ■ @Mn (ideal direct sum), then(ii) °R=°Mx®°M2® ■ ■ ■ ®°Mn (group direct product). Furthermore, (iii) \°R\ =11? \°Mi\, and hence |°P| <=o if and only if \°M,\ < oo for each i. Lemma 2 (Satz 4 of [8]). Every artinian ring is the direct sum of its torsion ideal and a torsionfree ideal whose additive group is divisible. Throughout the paper °R denotes the quasi-regular group of the ring, where the operation is taken to be a o b =a+b+ab, while J will be the Jacobson radical of R, and | M\ is the number of elements in the set M. If R has a unity, then *R will denote its group of units, in which case we also have °R~*R.