Editing Polynomial ring (section)

=== Properties that pass from {{math|''R''}} to {{math|''R''[''X'']}} ===
In this section, {{mvar|R}} is a commutative ring, {{mvar|K}} is a field, {{mvar|X}} denotes a single indeterminate, and, as usual, <math>\mathbb Z</math> is the ring of integers. Here is the list of the main ring properties that remain true when passing from {{mvar|R}} to {{math|''R''[''X'']}}.

* If {{mvar|R}} is an [[integral domain]] then the same holds for {{math|''R''[''X'']}} (since the leading coefficient of a product of polynomials is, if not zero, the product of the leading coefficients of the factors).
**In particular, <math>K[X_1,\ldots,X_n]</math> and <math>\mathbb Z[X_1,\ldots,X_n]</math> are integral domains.
* If {{mvar|R}} is a [[unique factorization domain]] then the same holds for {{math|''R''[''X'']}}. This results from [[Gauss's lemma (polynomial)|Gauss's lemma]] and the unique factorization property of <math>L[X],</math> where {{mvar|L}} is the field of fractions of {{mvar|R}}.
**In particular, <math>K[X_1,\ldots,X_n]</math> and <math>\mathbb Z[X_1,\ldots,X_n]</math> are unique factorization domains.
* If {{mvar|R}} is a [[Noetherian ring]], then the same holds for {{math|''R''[''X'']}}.
**In particular, <math>K[X_1,\ldots,X_n]</math> and <math>\mathbb Z[X_1,\ldots,X_n]</math> are Noetherian rings; this is [[Hilbert's basis theorem]].
* If {{mvar|R}} is a Noetherian ring, then <math>\dim R[X] = 1+\dim R,</math> where "<math>\dim</math>" denotes the [[Krull dimension]].
**In particular, <math>\dim K[X_1,\ldots,X_n] = n</math> and <math>\dim \mathbb Z[X_1,\ldots,X_n] = n+1.</math>
* If {{mvar|R}} is a [[regular ring]], then the same holds for {{math|''R''[''X'']}}; in this case, one has <math display="block">\operatorname{gl}\, \dim R[X]= \dim R[X]= 1 + \operatorname{gl}\, \dim R=1+\dim R,</math> where "<math>\operatorname{gl}\, \dim</math>" denotes the [[global dimension]].
**In particular, <math>K[X_1,\ldots,X_n]</math> and <math>\mathbb Z[X_1,\ldots,X_n]</math>  are regular rings, <math>\operatorname{gl}\, \dim \mathbb Z[X_1,\ldots,X_n] = n+1,</math> and <math>\operatorname{gl}\, \dim K[X_1,\ldots,X_n] = n.</math> The latter equality is [[Hilbert's syzygy theorem]].