A class of bounded hereditary Noetherian domains (Q759819)

Let k be a commutative field and F,G skew fields both containing k as a subfield of their centres. Given an (F,G)-bimodule M generated, as bimodule, by m, the authors define a k-algebra R(m) as follows: I(m) is the ideal of the ring coproduct (free product) \(P=F*_ kG\) generated by all elements \(\sum f_ ig_ i\) such that \(f_ i\in F\), \(g_ i\in G\) and \(\sum f_ img_ i=0\). Now \(R(m)=P/I(m)\). It was proved by \textit{A. H. Schofield} [Representations of rings over skew fields (1985), p.200] that R(m) is a fir. The authors use their structure theory of the representations of the bimodule \({}_ FM_ G\) to show that when F,G are finite dimensional over k, and \([M:F]=[M:G]=Z\), then R(m) is a bounded hereditary Noetherian domain, infinite-dimensional over k. As they remark, together with Schofield's results this shows R(m) to be a principal ideal domain.