The symmetric ring of quotients of the coproduct of rings (Q1178892)

Let \(K\) be a skew field and \(R_ 1,R_ 2\) \(K\)-rings whose left and right dimensions over \(K\) are greater than 2. The author proves that the coproduct \(R_ 1\coprod R_ 2\) is its own symmetric ring of quotients, unless i) \(R_ 1\) and \(R_ 2\) are primary, i.e. of the form \(K\oplus M\) where \(M\) is a \(K\)-bimodule such that \(M^ 2=0\) or ii) one factor is primary and the other is \(d\)-semiprimary or iii) each \(R_ i\) is \(d_ i\)-semiprimary. Here a \(d\)-semiprimary ring is a triangular matrix ring \(({K\atop V} {0\atop K})\), where \(V\) is a \(K\)-bimodule and the \((1,1)\) and \((2,2)\) entries are multiplied in the usual way, while the \((2,1)\) entries are multiplied by the rule \((\nu,\beta)(\gamma,w)^ T=\nu\gamma+\beta w+d(\beta)(\gamma-\delta)\) (where \(\delta\) is the \((2,2)\) entry of the second factor and \(d: K\to V\) is a derivation). This improves results of \textit{V. K. Kharchenko} [Algebra Logika 17, 478-487 (1978; Zbl 0433.16004)] and the author [Trans. Am. Math. Soc. 293, 303- 317 (1986; Zbl 0589.16003)]. Examples are given to show that each of the exceptions actually occurs.