Editing Ring (mathematics) (section)

=== Quotient ring ===
{{main|Quotient ring}}
The notion of [[quotient ring]] is analogous to the notion of a [[quotient group]].  Given a ring {{math|(''R'', +, '''⋅''')}} and a two-sided [[Ideal (ring theory)|ideal]] {{mvar|I}} of {{math|(''R'', +, '''⋅''')}}, view {{mvar|I}} as subgroup of {{math|(''R'', +)}}; then the '''quotient ring''' {{math|''R'' / ''I''}} is the set of [[coset]]s of {{mvar|I}} together with the operations
: <math>\begin{align}
& (a+I)+(b+I) = (a+b)+I, \\
& (a+I)(b+I) = (ab)+I.
\end{align}</math>
for all {{math|''a'', ''b''}} in {{mvar|R}}.  The ring {{math|''R'' / ''I''}} is also called a '''factor ring'''.

As with a quotient group, there is a canonical homomorphism {{math|''p'' : ''R'' → ''R'' / ''I''}}, given by {{math|''x'' ↦ ''x'' + ''I''}}. It is surjective and satisfies the following universal property: 
* If {{math|''f'' : ''R'' → ''S''}} is a ring homomorphism such that {{math|1=''f''(''I'') = 0}}, then there is a unique homomorphism <math>\overline{f} : R/I \to S</math> such that <math>f = \overline{f} \circ p.</math> 
For any ring homomorphism {{math|''f'' : ''R'' → ''S''}}, invoking the universal property with {{math|1=''I'' = ker ''f''}} produces a homomorphism <math>\overline{f} : R / \ker f \to S</math> that gives an isomorphism from {{math|''R'' / ker ''f''}} to the image of {{mvar|f}}.
<!-- need more work: A subset of {{math|''R''}} and the quotient {{math|''R'' / ''I''}} are related in the following way. A subset of {{mvar|R}} is called a [[system of representatives]] of {{math|''R'' / ''I''}} if no two elements in the set belong to the same coset, that is, each element in the set represents a unique coset. It is said to be complete if the restriction of {{math|''R'' → ''R'' / ''I''}} to it is surjective. -->