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”, in 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, 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:

     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