A Note on Noncommutative Polynomials

1. Introduction. We shall say that an integral domain1 R satisfies condition (M) if any two nonzero elements of R have a nonzero common right multiple. In this note it is proved that if 5 is an extension of a ring R such that S is, roughly speaking, a noncommutative polynomial ring in one variable with R as a coefficient ring, and if R has the property (M), then 5 has property (M). In case R is a division ring, this result has been proved by Ore [5].2 From our result it follows that the property (M) is preserved under an arbitrary number of extensions of the type described. It was first proved by Ore [4] that the condition (M) is necessary and sufficient in order that an integral domain R have a uniquely determined right quotient division ring. Our method is applied to prove that the Birkhoff-Witt algebra [l; 6] of a solvable Lie algebra over an arbitrary field of characteristic zero satisfies condition (M), and consequently has a uniquely determined right quotient division ring.3 It seems to be an unsolved problem to determine whether or not the Birkhoff-Witt algebra of an arbitrary Lie algebra satisfies condition (M). 2. Ring extensions. Let R and 5 be rings such that RQS. We shall say that S is an extension of type O of R if the following conditions are satisfied : (a) R and S have the same identity element. (b) S contains an element x not in R such that for each rÇ_R,