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Relationship of certain rings of infinite matrices over integers. (English) Zbl 0965.15014

Let \(\mathbb{N}\) be the set of nonnegative integers. Let \({\mathcal B}\) be the ring of \(\mathbb{N}\times \mathbb{N}\) matrices over the ring \(\mathbb{Z}\) of integers generated by two elements: one obtained by shifting the ones of the identity matrix one position to the right, call it \(X\), and the other one one position down, call it \(Y\). Let \({\mathcal C}\) be the subring of \({\mathcal B}\) generated by \(P= 1-X\) and \(Q= 1-Y\). The ring \({\mathcal F}\) of all \(\mathbb{N}\times \mathbb{N}\) matrices over \(\mathbb{Z}\) with only a finite number of nonzero entries and the ring \({\mathcal A}\) of all \(\mathbb{N}\times \mathbb{N}\) matrices over \(\mathbb{Z}\) with only a finite number of nonzero entries in each row and each column are also used in the paper. We have \[ {\mathcal F}\subset {\mathcal C}\subset {\mathcal B}\subset {\mathcal A}, \] and, obviously, \({\mathcal F}\) is an ideal of \({\mathcal A}\) (and thus of both \({\mathcal B}\) and \({\mathcal C}\)).
The paper deals with: essential extensions of rings with trivial annihilator, the usual embedding of a ring into a unitary one (here called the Dorroh extension), the mutual relationship of the rings \({\mathcal A}\) and \({\mathcal F}\), with some statements concerning all nonzero ideals of \({\mathcal A}\), and the mutual relationship between rings \({\mathcal B}\) and \({\mathcal C}\).

MSC:

15A30 Algebraic systems of matrices
15B36 Matrices of integers
16S70 Extensions of associative rings by ideals
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