The structure of rings of quotients

For an arbitrary ring R we completely characterize when Q(R), the maximal right ring of quotients of R, is a direct product of indecomposable rings and when Q(R) is a direct product of prime rings in terms of conditions on ideals of R. Our work generalizes decomposition results of Goodearl for a von Neumann regular right self-injective ring and of Jain, Lam, and Leroy for Q(R) when R is right nonsingular. To develop our results, we define a useful dimension on bimodules and characterize the subset of ideals of R which are dense in ring direct summands of Q(R). A structure theorem for RB(Q(R)), the subring of Q(R) generated by {re|r∈Rande∈B(Q(R))}, is provided for a semiprime ring R. Our methods allow us to properly generalize Rowen's theorem for semiprime PI-rings. We also apply our results to Functional Analysis to obtain a direct product decomposition of the local multiplier algebra, Mloc(A), of a C∗-algebra A. As a byproduct, we obtain a complete description of a C∗-algebra whose extended centroid is Cℵ. As a consequence, we show that a C∗-algebra with only finitely many minimal prime ideals and satisfying a polynomial identity is finite-dimensional.