Finite-dimensional right duo algebras are duo

Author:
R. C. Courter

Journal:
Proc. Amer. Math. Soc. **84** (1982), 157-161

MSC:
Primary 16A15; Secondary 16A46

DOI:
https://doi.org/10.1090/S0002-9939-1982-0637159-6

MathSciNet review:
637159

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Abstract: Available examples of right (but not left) duo rings include rings without unity element which are two dimensional algebras over finite prime fields. We prove that a right duo ring with unity element which is a finite dimensional algebra over an arbitrary field is a duo ring. This result is obtained as a corollary of a theorem on right duo, right artinian rings with unity: left duo-ness is equivalent to each right ideal of having equal right and left composition lengths, which is equivalent to the same property on alone. Another result concerns algebras over a field which are semiprimary right duo rings: such an algebra is left duo provided (1) the algebra is finite dimensional modulo its radical and (2) the square of the radical is zero. These two provisions are shown to be essential by examples which are local algebras, duo on one side only.

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DOI:
https://doi.org/10.1090/S0002-9939-1982-0637159-6

Article copyright:
© Copyright 1982
American Mathematical Society