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On directly infinite rings. (English) Zbl 0991.16024

Let \(R\) be a ring with 1. \(R\) is called directly finite if one-sided units are (two-sided) units in \(R\); in the contrary case \(R\) is called directly infinite. The connection between directly infinite rings and infinite matrix rings was noted by N. Jacobson [Proc. Am. Math. Soc. 1, 352-355 (1950; Zbl 0037.15901)]. These concepts extend naturally to semigroups. Let \(C=\mathbb{N}\times\mathbb{N}\), \(\mathbb{N}\) the set of nonnegative integers, \((m,n)(p,q)=(m+p-r,n+q-r)\) with \(r=\min\{n,p\}\). Then \(C\) is a monoid called the bicyclic semigroup. \(R\) is called bicyclic if it is generated by a bicyclic submonoid of its multiplicative semigroup. In a monoid \(ab=1\neq ba\) if and only if the mapping \(a\mapsto(0,1)\), \(b\mapsto(1,0)\) extends to an isomorphism between the submonoid generated by \(a\) and \(b\) and the bicyclic semigroup \(C\). The aim of the paper is to give ring theoretical analogues of this result.
Let \(\mathcal A\) be the ring of infinite matrices over the ring of integers with only finitely many nonzero entries in each row and column, and for \(m,n\in\mathbb{N}\) let \(\langle m,n\rangle\) denote the matrix in \(\mathcal A\) with 1 at the positions \((m+t,n+t)\), \(t=0,1,2,\dots\) and zeros elsewhere. Let \(\mathcal B\) (\(B\)) be the subring (subsemigroup) of (the multiplicative semigroup of) \(\mathcal A\) generated by \(\langle 0,1\rangle\) and \(\langle 1,0\rangle\). The property \(\langle m,n\rangle\langle p,q\rangle=\langle m+p-r,n+q-r\rangle\) connects bicyclic semigroups and the ring \(\mathcal B\).
The principal result of the paper is as follows. For \(a,b\in R\) define the function \(\varphi\colon\langle 0,1\rangle\mapsto a\), \(\langle 1,0\rangle\mapsto b\). Then the following are equivalent: 1. \(ab=1\neq ba\); 2. \(\varphi\) extends uniquely to an isomorphism between \(B\) and the subsemigroup generated by \(a\) and \(b\); 3. \(\varphi\) extends uniquely to an isomorphism between \(\mathcal B\) and the subring generated by \(a,b\) and 1. As a consequence, \(R\) is directly infinite if and only if it has a bicyclic subring with 1. The paper also studies idempotent elements.

MSC:

16S50 Endomorphism rings; matrix rings
20M25 Semigroup rings, multiplicative semigroups of rings
20M10 General structure theory for semigroups
16U60 Units, groups of units (associative rings and algebras)

Citations:

Zbl 0037.15901
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