Regular sequence on a module: Difference between revisions
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* <math>(x_1,x_2,\ldots,x_n)M \ne M</math> | * <math>(x_1,x_2,\ldots,x_n)M \ne M</math> | ||
* For <math>1 \le i \le n</math>, <math>x_i</math> is a nonzerodivisor on <math>M/(x_1,x_2,\ldots,x_{i-1})</math> | * For <math>1 \le i \le n</math>, <math>x_i</math> is a [[nonzerodivisor on a module|nonzerodivisor]] on <math>M/(x_1,x_2,\ldots,x_{i-1})</math> | ||
==Facts== | ==Facts== | ||
If <math>R</math> is a [[Noetherian ring|Noetherian]] [[local ring]] and <math>x_1, x_2, \ldots, x_n</math> form a regular sequence in its unique [[maximal ideal]], then any permutation of the <math>x_i</math>s also forms a regular sequence in the maximal ideal. | If <math>R</math> is a [[Noetherian ring|Noetherian]] [[local ring]] and <math>x_1, x_2, \ldots, x_n</math> form a regular sequence in its unique [[maximal ideal]], then any permutation of the <math>x_i</math>s also forms a regular sequence in the maximal ideal. | ||
Revision as of 21:37, 16 March 2008
Definition
Let be a commutative unital ring, a -module, and be a sequence of elements in . We say that the s form a regular sequence on if the following two conditions hold:
- For , is a nonzerodivisor on
Facts
If is a Noetherian local ring and form a regular sequence in its unique maximal ideal, then any permutation of the s also forms a regular sequence in the maximal ideal.