A canonical directly infinite ring.

*(English)*Zbl 1079.15508Summary: Let \(\mathbb N\) be the set of nonnegative integers and \(\mathbb Z\) the ring of integers. Let \(\mathcal B\) be the ring of \(\mathbb N \times \mathbb N\) 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.

PDF
BibTeX
XML
Cite

\textit{M. Petrich} and \textit{P. V. Silva}, Czech. Math. J. 51, No. 3, 545--560 (2001; Zbl 1079.15508)

**OpenURL**

##### References:

[1] | K. R. Goodearl: Von Neumann Regular Rings. Pitman, London, 1979. · Zbl 0411.16007 |

[2] | N. Jacobson: Lectures in Abstract Algebra II. Linear algebra. Springer, New York-Heidelberg-Berlin, 1975. · Zbl 0314.15001 |

[3] | S. Lang: Algebra. Addison-Wesley, Reading, 1993. · Zbl 0848.13001 |

[4] | M. Petrich, P. V. Silva: On directly infinite rings. Acta Math. Hungar. 85 (1999), 153-165. · Zbl 0991.16024 |

[5] | M. Petrich, P. V. Silva: On presentations of semigroup rings. Boll. Un. Mat. Ital. B 8 (1999), 127-142. · Zbl 0924.16022 |

[6] | L. H. Rowen: Ring Theory, Vol. I. Academic Press, San Diego, 1988. · Zbl 0651.16002 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.