Abstract. A ring $R$ is said to be quasi directly finite if for any $a,b \in R$ such that $a+b = ab$ we have $ab = ba$; otherwise $R$ is quasi directly infinite. In order to study the latter rings, we introduce a ring ${\cal C}$ of $N \times N$ matrices over $Z$ generated by two elements, where $N$ is the set of nonnegative integers and $Z$ is the ring of integers. We characterize the ring ${\cal C}$ in several ways including the fact that ${\cal C}$ is isomorphic to the augmentation ideal of a semigroup ring. The main result shows that a ring $R$ is quasi directly infinite if and only if it contains a certain homomorphic image of the ring ${\cal C}$. Several ramifications of this result provide further characterizations of such rings as well as their relationship with directly infinite rings.

AMS Subject Classification (1991): 16P99, 15A36, 20M25

Received November 21, 1997 and in revised form October 12, 1998. (Registered under 2672/2009.)