Generalized Quotient Rings

Let D be a domain with quotient field K . In this thesis, we consider "generalized'1 quotient rings of D obtained by replacing a multiplicative system of elements of D by a multiplicative system S of non-zero subsetsof D . We call the ring = {x e k|there exists A e S such that xA c d ) a Generalized Quotient Ring (GQR) of D and we call S a Generalized Multiplicative System (GMS) of subsets of D . If Q is a ring containing D and there exists a GMS of subsets of S of D such that Q = and A e S implies AQ = Q , then Q is said to be a Restricted GQR (RGQR) of D . It is the purpose of this thesis to study GQR1s of D . It is clear that ideal transforms introduced by Nagata in [N.2] are such rings. It is also true that ordinary quotient rings, intersections of localizations and flat overrings (as studied by Richman in [R] and Akiba in [A.l] and [A.2]) of D are such rings. If R is a commutative ring with identity that is not necessarily a domain, then we may define a GQR of R in much that same manner as we define an ordinary quotient ring of R . In the first chapter, we give this definition,