Editing Integral domain (section)

== Properties ==
* A commutative ring ''R'' is an integral domain if and only if the ideal (0) of ''R'' is a prime ideal.
* If ''R'' is a commutative ring and ''P'' is an [[ideal (ring theory)|ideal]] in ''R'', then the [[quotient ring]] ''R/P'' is an integral domain if and only if ''P'' is a [[prime ideal]].
* Let ''R'' be an integral domain. Then the [[polynomial ring]]s over ''R'' (in any number of indeterminates) are integral domains. This is in particular the case if ''R'' is a [[field (mathematics)|field]].
* The cancellation property holds in any integral domain: for any ''a'', ''b'', and ''c'' in an integral domain, if {{nowrap|''a'' ≠ ''0''}} and {{nowrap|1=''ab'' = ''ac''}} then {{nowrap|1=''b'' = ''c''}}. Another way to state this is that the function {{nowrap|''x'' ↦ ''ax''}} is injective for any nonzero ''a'' in the domain.
* The cancellation property holds for ideals in any integral domain: if {{nowrap|1=''xI'' = ''xJ''}}, then either ''x'' is zero or {{nowrap|1=''I'' = ''J''}}.
* An integral domain is equal to the intersection of its [[localization of a ring|localizations]] at maximal ideals.
* An [[inductive limit]] of integral domains is an integral domain.
* If ''A'', ''B'' are integral domains over an algebraically closed field ''k'', then {{nowrap|''A'' ⊗<sub>''k''</sub> ''B''}} is an integral domain. This is a consequence of [[Hilbert's nullstellensatz]],{{efn|Proof: First assume ''A'' is finitely generated as a ''k''-algebra and pick a ''k''-basis <math>g_i</math> of ''B''. Suppose <math display="inline">\sum f_i \otimes g_i \sum h_j \otimes g_j = 0</math> (only finitely many <math>f_i, h_j</math> are nonzero). For each maximal ideal <math>\mathfrak{m}</math> of ''A'', consider the ring homomorphism <math>A \otimes_k B \to A/\mathfrak{m} \otimes_k B = k \otimes_k B \simeq B</math>. Then the image is <math display="inline">\sum \overline{f_i} g_i \sum \overline{h_i} g_i = 0</math> and thus either <math display="inline">\sum \overline{f_i} g_i = 0</math> or <math display="inline">\sum \overline{h_i} g_i = 0</math> and, by linear independence, <math>\overline{f_i} = 0</math> for all <math>i</math> or <math>\overline{h_i} = 0</math> for all <math>i</math>. Since <math>\mathfrak{m}</math> is arbitrary, we have <math display="inline">(\sum f_iA) (\sum h_iA) \subset \operatorname{Jac}(A) = </math> the intersection of all maximal ideals <math>= (0)</math> where the last equality is by the Nullstellensatz. Since <math>(0)</math> is a prime ideal, this implies either <math display="inline">\sum f_iA</math> or <math display="inline">\sum h_iA</math> is the zero ideal; i.e., either <math>f_i</math> are all zero or <math>h_i</math> are all zero. Finally, ''A'' is an inductive limit of finitely generated ''k''-algebras that are integral domains and thus, using the previous property, <math>A \otimes_k B = \varinjlim A_i \otimes_k B</math> is an integral domain. <math>\square</math>}} and, in algebraic geometry, it implies the statement that the coordinate ring of the product of two affine algebraic varieties over an algebraically closed field is again an integral domain.