Dictionary

quotient ring en · NOUN

Meanings

1. (qualifier: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 maximal right and left quotient rings. 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:
   • 2. An ideal P of the ring R is prime if and only if the quotient ring R/P is a domain. 2012, Oleg A. Logachev, A. A. Salnikov, V. V. Yashchenko, translated by Svetla Nikova, Boolean Functions in Coding Theory and Cryptography, American Mathematical Society, page 10:

Forms

Deriveds

maximal quotient ring · right quotient ring · left quotient ring

Hyponyms

residue field

Relateds

quotient module · quotient space · quotient object · quotient group

Synonyms

factor ring (ring whose elements are the cosets of an ideal) · residue class ring (ring whose elements are the cosets of an ideal) · difference ring (ring whose elements are the cosets of an ideal)

Translations (1)

fr anneau quotient (ring whose elements are the cosets of an ideal)