quotient ring

quotient ring

English

Noun

quotient ring (plural quotient rings)

1. (algebra, 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.
   • 1976, Kenneth Goodearl, Ring Theory: Nonsingular Rings and Modules, CRC Press, page 39,

     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.

   • 2006, Peter A. Linnell, Noncommutative localization in group rings, Andrew Ranicki (editor), Noncommutative Localization in Algebra and Topology, Cambridge University Press, page 42,

     On the other hand if already every element of R is either invertible or a zerodivisor, then R is its own classical quotient ring.

   • 2012, Svetla Nikova (translator), Oleg A. Logachev, A. A. Salnikov, V. V. Yashchenko, Boolean Functions in Coding Theory and Cryptography, American Mathematical Society, page 10,

     2. An ideal P of the ring R is prime if and only if the quotient ring R/P is a domain.

Synonyms

• (ring whose elements are the cosets of an ideal): difference ring, factor ring, residue class ring

Hyponyms

• residue field

Derived terms

• left quotient ring
• right quotient ring
• maximal quotient ring

Translations

ring whose elements are the cosets of an ideal

French: anneau quotient m

See also

• quotient group
• quotient module
• quotient object
• quotient space

Further reading

• Quotient module on Wikipedia.Wikipedia
• Residue field on Wikipedia.Wikipedia