Geodesic
A canonical directly infinite ring
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 3, pp. 545-560 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $\mathbb N$ be the set of nonnegative integers and $\mathbb Z$ the ring of integers. Let $\mathcal B$ be the ring of $N \times N$ matrices over $\mathbb Z$ 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.
Let $\mathbb N$ be the set of nonnegative integers and $\mathbb Z$ the ring of integers. Let $\mathcal B$ be the ring of $N \times N$ matrices over $\mathbb Z$ 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.
Classification : 15A36, 16D60, 16P70, 16S50, 16U60
Keywords: directly finite rings; matrix rings 
@article{CMJ_2001_51_3_a7,
     author = {Petrich, Mario and Silva, Pedro V.},
     title = {A canonical directly infinite ring},
     journal = {Czechoslovak Mathematical Journal},
     pages = {545--560},
     year = {2001},
     volume = {51},
     number = {3},
     mrnumber = {1851546},
     zbl = {1079.15508},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a7/}
}
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Petrich, Mario; Silva, Pedro V. A canonical directly infinite ring. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 3, pp. 545-560. http://geodesic.mathdoc.fr/item/CMJ_2001_51_3_a7/