Classical quotient rings of trivial extensions
Let \(M\) be an \(R\)-\(R\)-bimodule and let \(L\) be the trivial extension whose elements (with an obvious notation) are of the form \((r,m)\) with componentwise addition and \((r,m)(r',m') = (rr',rm'+mr')\). Let \(D\) be the set of regular elements \(c\) of \(R\) such that cm\(\neq 0\neq mc\) for every nonzero element \(m\) of \(M\). Suppose either that \(M\) is faithful or that \(M\) has finite Goldie dimension on both sides. It is shown that an element \((r,m)\) of \(L\) is regular if and only if \(r\) is in \(D\). Also \(L\) has a classical right quotient ring if and only if both \(R\) and \(M\) satisfy the right Ore condition with respect to \(D\), and the right quotient ring of \(L\) is itself a trivial extension in the natural way.
scientific article; zbMATH DE number 3970959