A Canonical Directly Infinite Ring

Abstract

Let ℕ be the set of nonnegative integers and ℤ the ring of integers. Let $\mathcal{B}$ be the ring of N × N matrices over ℤ generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of $\mathcal{B}$ yields that the subrings generated by them coincide. This subring is the sum of the ideal $\mathcal{F}$ consisting of all matrices in $\mathcal{B}$ with only a finite number of nonzero entries and the subring of $\mathcal{B}$ generated by the identity matrix. Regular elements are also described. We characterize all ideals of $\mathcal{B}$, show that all ideals are finitely generated and that not all ideals of $\mathcal{B}$ are principal. Some general ring theoretic properties of $\mathcal{B}$ are also established.