n. 1. another term for factor ring. 2. an over-ring, Q, of a given ring, R, with some regular elements, that consists of elements of the form ab^-1 (in a right quotient ring) or b^-1a (in a left quotient ring) where a is any element and b is a regular element of R, and the inverses of all regular elements of R are in Q. Any semi-prime right Noetherian ring has a semi-simple right quotient ring, and any integral domain has a quotient ring that is a field, the field of fractions.