Even when adding "over the real number", the claim in the article is wrong: R(x), the algebra of rational functions of one variable with real coefficients is a

Even when adding "over the real number", the claim in the article is wrong: R(x), the algebra of rational functions of one variable with real coefficients is a field, and a module over R containing a isomorphic image of R as a subfield. You really do need the additional constraint of the Hurwitz form to restrict the possibilities to R, C, H, O ... (Of course this has nothing to do with the work of Furey - I'm just seconding that the claim in the article is incomplete and inexact as worded.)