# A canonical directly infinite ring

• Volume: 51, Issue: 3, page 545-560
• ISSN: 0011-4642

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## Abstract

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Let $ℕ$ be the set of nonnegative integers and $ℤ$ the ring of integers. Let $ℬ$ 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 $ℬ$ yields that the subrings generated by them coincide. This subring is the sum of the ideal $ℱ$ consisting of all matrices in $ℬ$ with only a finite number of nonzero entries and the subring of $ℬ$ generated by the identity matrix. Regular elements are also described. We characterize all ideals of $ℬ$, show that all ideals are finitely generated and that not all ideals of $ℬ$ are principal. Some general ring theoretic properties of $ℬ$ are also established.

## How to cite

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Petrich, Mario, and Silva, Pedro V.. "A canonical directly infinite ring." Czechoslovak Mathematical Journal 51.3 (2001): 545-560. <http://eudml.org/doc/30654>.

@article{Petrich2001,
abstract = {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.},
author = {Petrich, Mario, Silva, Pedro V.},
journal = {Czechoslovak Mathematical Journal},
keywords = {directly finite rings; matrix rings; directly finite rings; matrix rings},
language = {eng},
number = {3},
pages = {545-560},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A canonical directly infinite ring},
url = {http://eudml.org/doc/30654},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Petrich, Mario
AU - Silva, Pedro V.
TI - A canonical directly infinite ring
JO - Czechoslovak Mathematical Journal
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 3
SP - 545
EP - 560
AB - 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.
LA - eng
KW - directly finite rings; matrix rings; directly finite rings; matrix rings
UR - http://eudml.org/doc/30654
ER -

## References

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1. Von Neumann Regular Rings, Pitman, London, 1979. (1979) Zbl0411.16007MR0533669
2. Lectures in Abstract Algebra II. Linear algebra, Springer, New York-Heidelberg-Berlin, 1975. (1975) Zbl0314.15001MR0392906
4. 10.1023/A:1006633231817, Acta Math. Hungar. 85 (1999), 153–165. (1999) MR1713097DOI10.1023/A:1006633231817
5. On presentations of semigroup rings, Boll. Un. Mat. Ital. B 8 (1999), 127–142. (1999) MR1794554
6. Ring Theory, Vol. I, Academic Press, San Diego, 1988. (1988) MR0940245

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