Endomorphism rings of bimodules. (Q2517643)

For an integral domain \(R\), the endomorphism ring of a torsion-free \(R\)-module is torsion-free, as well. The authors investigate if this remains true for non-commutative rings, where ``torsion-free'' is replaced by ``non-singular''. Let \(R\) be a unital but not necessarily commutative ring. A right module \(M\) over \(R\) is right non-singular if \(xI\neq 0\) for every essential right ideal \(I\) and \(0\neq x\in M\). The ring \(R\) is right non-singular if \(R_R\) is right non-singular (left non-singularity is defined similarly). Further, an \(R\)-\(R\)-bimodule \(M\) such that \(M_R\) and \(_RM\) are non-singular has the right essentiality property if \(IM_R\) is essential in \(M_R\) for all essential right ideals \(I\) of \(R\). Now, if an \(R\)-\(R\)-bimodule \(M\) such that \(M_R\) and \(_R M\) are non-singular has the right essentiality property then the endomorphism ring of \(M\) is non-singular. Two main results of the paper are the following. Theorem: [part of Theorem~2.4.] Let \(R\) be a ring with maximal right and left ring of quotients \(Q(R)\). Then every \(R\)-\(R\)-bimodule \(M\) such that \(_RM\) and \(M_R\) are non-singular and finite dimensional (i.e.\ contain no infinite direct sums of non-zero submodules) has the right essentiality property if and only if \(R\) is semi-prime. Further, for such a ring, \(R^X\) has the right essentiality property for all index-sets \(X\) if and only if \(R\) is right bounded, that is, every essential right ideal of \(R\) contains a two-sided ideal which is essential as a right ideal. Theorem: [part of Theorem~3.1.] Let \(R\) be a semi-prime, right and left bounded right and left Goldie ring for which \(R^+\) is torsion-free. Then every \(R\)-\(R\)-bimodule \(M\) such that \(_RM\) and \(M_R\) are non-singular has the right essentiality property if and only if \(\mathbb Q\otimes_{\mathbb Z}R\) is semi-simple Artinian.