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Rings of Stable Rank 2 are Barbilian Rings

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Let MR a free unitary modul. A non-empty subset B of MR is by definition a Barbilian set iff B* ≔ F ⊂ B ¦ F is a free generating system of MR and #F > 1 satisfies: (B1) Each u1 ϵ B may be completed to 1, u2, … ϵ B* (B1) 1, u2, … ϵ B* implies 1 + u2α, u2, … ϵ B* for all α ϵ R. For each Barbilian set B ⊂ MR holds: B ⊂ Bmax = ϵ MR ¦ u may be completed to a free generating system F of MR and #F > 1 and Bmax is a Barbilian set iff Bmax ≠ .

A ring R is by definition a Barbilian ring iff each Barbilian set B over any free unitary R-module MR coincides with Bmax. We show that all rings of stable rank 2 are Barbilian rings. The class of rings of stable rank 2 covers for instance the class of all semiprimary rings which contains the class of all finite rings with identity. This answers two questions of W. BENZ, posed a few years ago to the author.

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References

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Authors and Affiliations

  1. Werner Leißner Fachbereich 6 der Universität Oldenburg, Ammerländer Heerstr. 114-118, W-2900, Oldenburg

    Werner Leißner

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  1. Werner Leißner
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Dedicated to my highly esteemed teacher WALTER BENZ on occasion of his 60th birthday

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Leißner, W. Rings of Stable Rank 2 are Barbilian Rings. Results. Math. 20, 530–537 (1991). https://doi.org/10.1007/BF03323191

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  • DOI: https://doi.org/10.1007/BF03323191

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