Finite-dimensional right duo algebras are duo

R. C. Courter
1982 Proceedings of the American Mathematical Society  
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 A' is a duo ring. This result is obtained as a corollary of a theorem on right duo, right artinian rings R with unity: left duo-ness is equivalent to each right ideal of R having equal right and left composition lengths, which is
more » ... s, which is equivalent to the same property on R 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. Proposition 1.1. If R is a right duo ring, so is each of its homomorphic images. Remark 1.2. An invertible element x of a ring R satisfies xR = Rx: if r E R, rx = xx~xrx E xR. Theorem 1.3. Let e be an idempotent element of a right duo ring R. Then e belongs to the center of R. Consequently, right artinian, right duo simple rings are division rings. Proof. Let/ = 1 -e. If e is not a central element, there exists an element x E R satisfying either exf =£ 0 orfxe ¥= 0. If exf ¥= 0, the right duo-ness of R implies that exfEfR. But eR n fR = 0, contradicting exf ^ 0. Thus each idempotent is central. The second conclusion is obvious now. Examples. In [5, p. 149, Theorem 1] it is proved that a skew power series ring D(t, a}, defined over a commutative principal ideal domain D with the aid of a
doi:10.1090/s0002-9939-1982-0637159-6 fatcat:52imz2vqybgybjnl4sa4k67mpa