Quotient rings of a ring and a subring which have a common right ideal

Abstract:

Let R be a subring of S and let $A \subseteq R$ be a right ideal of S. In this paper we show that there is a bijection between right torsion theories $\tau$ over S such that A is $\tau$-dense and right torsion theories $\sigma$ over R such that S/A is $\sigma$-torsion. A similar result is obtained for the left side with the bijection between torsion theories over S with SA dense and torsion theories over R with RA dense. It is also shown that the ring of quotients of R and S at these corresponding torsion theories are equal. As a corollary, when A is chosen appropriately R and S have the same right (left) maximal quotient ring.