quotient ring

Noun

quotient ring (plural quotient rings)

1. (ring theory) For a given ring R and ideal I contained in R, another ring, denoted R / I, whose elements are the cosets of I in R.
   • The third section covers a construct similar to the ring S°R — the maximal quotient ring, which exists for any ring. (When R is nonsingular, the maximal quotient ring is exactly S°R.) Finally, Section D provides an answer to the question of which right and left nonsingular rings have coinciding...1976, Kenneth Goodearl, Ring Theory: Nonsingular Rings and Modules, CRC Press, page 39
   • On the other hand if already every element of R is either invertible or a zerodivisor, then R is its own classical quotient ring.2006, Peter A. Linnell, “Noncommutative localization in group rings”, in Andrew Ranicki, editor, Noncommutative Localization in Algebra and Topology, Cambridge University Press, page 42

   Synonyms:

   difference ring, factor ring, residue class ring