The structure of rings of quotients.
Let \(R\) be an associative ring with identity, \(Q(R)\) be its maximal right quotient ring, \(\mathbf B(Q(R))\) be the set of central idempotents in \(Q(R)\), and \(R\mathbf B(Q(R))\) be the subring of \(Q(R)\) generated by \(\{re:r\in R,\;e\in\mathbf B(Q(R))\}\). This paper investigates the connections between the ideal structure of \(R\) and that of \(Q(R)\) and \(R\mathbf B(Q(R))\). In particular, necessary and sufficient conditions are obtained for \(Q(R)\) to be a direct product of indecomposable rings or a direct product of prime rings. In doing so, the authors remove the nonsingularity condition assumed by their predecessors. Their methods employ a new dimension on a bimodule \(M\), which they call the `Johnson dimension', defined to be the supremum of the integers \(n\) for which there is a direct sum of \(n\) nonzero bimodules which is dense in \(M\). If \(R\) is semiprime, it is shown that \(R\mathbf B(Q(R))\) is the smallest right ring of quotients of \(R\) having the property that every ideal is essential in a ring direct summand. In this case, \(R\) has exactly \(n\) minimal prime ideals \(P_1,\dots,P_n\) if and only if \(R\mathbf B(Q(R))\cong R/P_1\oplus\cdots\oplus R/P_n\). Moreover, they prove that if \(R\) is semiprime and each prime factor ring of \(R\) is a PI-ring, then every nonzero ideal of \(R\) has nonzero intersection with the centre of \(R\), thereby generalizing a well-known result of Rowen. The paper ends with a section on how the results can be applied to not necessarily unital \(C^*\)-algebras. For example, it is shown that if \(A\) is a \(C^*\)-algebra then its local multiplier algebra \(M_{\text{loc}}(A)\) is a \(C^*\)-algebra direct product of \(\aleph\) prime \(C^*\)-algebras if and only if the extended centroid of \(A\) is \(\mathbb{C}^\aleph\), where \(\aleph\) is a cardinality and \(\mathbb{C}\) is the complex field.
scientific article; zbMATH DE number 5572166
scientific article