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Rings isomorphic to \(3\times 3\) matrix rings. (English) Zbl 0793.16019

It is shown that a ring \(R\) with identity element is isomorphic to a full 3 by 3 matrix ring if and only if there are non-zero elements \(x\), \(y\), \(z\), \(a\) of \(R\) such that \(x^ 2 = 0 = z^ 2\), \(y^ 2 = y\), \((x + y + z)^ 2 = 1\), \(a = ay + ya = a(x + z) + (x + z)a\), and \(y\) annihilates both \(x\) and \(z\).

MSC:

16S50 Endomorphism rings; matrix rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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