quotient ring
noun1
Algebra, Mathematics, Sciences 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:
1 more example
- 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:
- Synonyms
- difference ring, factor ring, residue class ring
- Derived terms
- left quotient ring, right quotient ring, maximal quotient ring
- Related terms
- quotient group, quotient module, quotient object, quotient space
- Hyponyms
- residue field