Editing Quotient ring (section)

== Properties ==
Clearly, if <math>R</math> is a [[commutative ring]], then so is {{tmath|1= R\ /\ I }}; the converse, however, is not true in general.

The natural quotient map <math>p</math> has <math>I</math> as its [[kernel (algebra)|kernel]]; since the kernel of every ring homomorphism is a two-sided ideal, we can state that two-sided ideals are precisely the kernels of ring homomorphisms.

The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows: the ring homomorphisms defined on <math>R\ /\ I</math> are essentially the same as the ring homomorphisms defined on <math>R</math> that vanish (i.e. are zero) on {{tmath|1= I }}. More precisely, given a two-sided ideal <math>I</math> in <math>R</math> and a ring homomorphism <math>f : R \to S</math> whose kernel contains {{tmath|1= I }}, there exists precisely one ring homomorphism <math>g : R\ /\ I \to S</math> with <math>gp = f</math> (where <math>p</math> is the natural quotient map). The map <math>g</math> here is given by the well-defined rule <math>g([a]) = f(a)</math> for all <math>a</math> in {{tmath|1 R }}. Indeed, this [[universal property]] can be used to ''define'' quotient rings and their natural quotient maps.

As a consequence of the above, one obtains the fundamental statement: every ring homomorphism <math>f : R \to S</math> induces a [[ring isomorphism]] between the quotient ring <math>R\ /\ \ker (f)</math> and the image {{tmath|1= \mathrm{im} (f) }}. (See also: ''[[Fundamental theorem on homomorphisms]]''.)

The ideals of <math>R</math> and <math>R\ /\ I</math> are closely related: the natural quotient map provides a [[bijection]] between the two-sided ideals of <math>R</math> that contain <math>I</math> and the two-sided ideals of <math>R\ /\ I</math> (the same is true for left and for right ideals). This relationship between two-sided ideal extends to a relationship between the corresponding quotient rings: if <math>M</math> is a two-sided ideal in <math>R</math> that contains {{tmath|1= I }}, and we write <math>M\ /\ I</math> for the corresponding ideal in <math>R\ /\ I</math> (i.e. {{tmath|1= M\ /\ I = p(M) }}), the quotient rings <math>R\ /\ M</math> and <math>(R / I)\ /\ (M / I)</math> are naturally isomorphic via the (well-defined) mapping {{tmath|1= a + M \mapsto (a + I) + M / I }}.

The following facts prove useful in [[commutative algebra]] and [[algebraic geometry]]: for <math>R \neq \lbrace 0 \rbrace</math> commutative, <math>R\ /\ I</math> is a [[Field (mathematics)|field]] if and only if <math>I</math> is a [[maximal ideal]], while <math>R / I</math> is an [[integral domain]] if and only if <math>I</math> is a [[prime ideal]]. A number of similar statements relate properties of the ideal <math>I</math> to properties of the quotient ring {{tmath|1= R\ /\ I }}.

The [[Chinese remainder theorem]] states that, if the ideal <math>I</math> is the intersection (or equivalently, the product) of pairwise [[coprime#Generalizations|coprime]] ideals {{tmath|1= I_1, \ldots, I_k }}, then the quotient ring <math>R\ /\ I</math> is isomorphic to the [[product of rings|product]] of the quotient rings {{tmath|1= R\ /\ I_n,\; n = 1, \ldots, k }}.