On the Dedekind-finite rings. Applications to linear algebra (Q2706761)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the Dedekind-finite rings. Applications to linear algebra |
scientific article |
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25 March 2001
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Dedekind-finite rings
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rings of formal power series
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triangular matrix rings
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polynomial rings
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units
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On the Dedekind-finite rings. Applications to linear algebra (English)
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A unitary ring \(R\) is called Dedekind-finite if for all elements \(a,b\in R\) satisfying \(ab=1\) one also has \(ba=1\). The authors prove that if \(R\) is Dedekind-finite, then the ring \(R[[X]]\) of formal power series and in particular the ring \(R[X]\) are Dedekind-finite. As an application, a generalization of the fundamental theorem of similar matrices is obtained. Another result tells that the ring \(T_n(R)\) of triangular matrices over a Dedekind-finite ring \(R\) is also Dedekind-finite.
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