Editing Quotient ring (section)

{{short description|Reduction of a ring by one of its ideals}}
{{Ring theory sidebar|Basic}}

In [[ring theory]], a branch of [[abstract algebra]], a '''quotient ring''', also known as '''factor ring''', '''difference ring'''<ref>{{cite book | author-link=Nathan Jacobson | last1=Jacobson | first1=Nathan | title=Structure of Rings | publisher=American Mathematical Soc. | year=1984 | edition=revised | isbn=0-821-87470-5}}</ref> or '''residue class ring''', is a construction quite similar to the [[quotient group]] in [[group theory]] and to the [[quotient space (linear algebra)|quotient space]] in [[linear algebra]].<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge | author-link=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref> It is a specific example of a [[quotient (universal algebra)|quotient]], as viewed from the general setting of [[universal algebra]]. Starting with a [[ring (mathematics)|ring]] <math>R</math> and a [[two-sided ideal]] <math>I</math> in {{tmath|1= R }}, a new ring, the quotient ring {{tmath|1= R\ /\ I }}, is constructed, whose elements are the [[cosets]] of <math>I</math> in <math>R</math> subject to special <math>+</math> and <math>\cdot</math> operations. (Quotient ring notation always uses a [[fraction slash]] "{{tmath|1= / }}".)

Quotient rings are distinct from the so-called "quotient field", or [[field of fractions]], of an [[integral domain]] as well as from the more general "rings of quotients" obtained by [[localization of a ring|localization]].