Editing Quotient ring (section)

== Formal quotient ring construction ==
Given a ring <math>R</math> and a two-sided ideal <math>I</math> in {{tmath|1= R }}, we may define an [[equivalence relation]] <math>\sim</math> on <math>R</math> as follows:
: <math>a \sim b</math> [[if and only if]] <math>a - b</math> is in {{tmath|1= I }}.
Using the ideal properties, it is not difficult to check that <math>\sim</math> is a [[congruence relation]].
In case {{tmath|1= a \sim b }}, we say that <math>a</math> and <math>b</math> are ''congruent [[Ideal (ring theory)|modulo]]'' <math>I</math> (for example, <math>1</math> and <math>3</math> are congruent modulo <math>2</math> as their difference is an element of the ideal {{tmath|1= 2 \mathbb{Z} }}, the [[Parity (mathematics)|even integers]]). The [[equivalence class]] of the element <math>a</math> in <math>R</math> is given by:
<math display="block">\left[ a \right] = a + I := \left\lbrace a + r : r \in I \right\rbrace</math>

This equivalence class is also sometimes written as <math>a \bmod I</math> and called the "residue class of <math>a</math> modulo <math>I</math>".

The set of all such equivalence classes is denoted by {{tmath|1= R\ /\ I }}; it becomes a ring, the '''factor ring''' or '''quotient ring''' of <math>R</math> modulo {{tmath|1 = I }}, if one defines
* {{tmath|1= (a + I) + (b + I) = (a + b) + I }};
* {{tmath|1= (a + I)(b + I) = (ab) + I }}.
(Here one has to check that these definitions are [[well-defined]]. Compare [[coset]] and [[quotient group]].) The zero-element of <math>R\ /\ I</math> is {{tmath|1= \bar{0} = (0 + I) = I }}, and the multiplicative identity is {{tmath|1= \bar{1} = (1 + I) }}.

The map <math>p</math> from <math>R</math> to <math>R\ /\ I</math> defined by <math>p(a) = a + I</math> is a [[surjective]] [[ring homomorphism]], sometimes called the '''''natural quotient map''''' or the '''''[[canonical homomorphism]]'''''.