Regular sequence in a ring: Difference between revisions

Latest revision as of 16:34, 12 May 2008

Definition

Suppose $R$ is a commutative unital ring, and $x_{1}, x_{2}, \dots, x_{n}$ is a sequence of elements in $R$ . We say that the $x_{i}$ s form a regular sequence in $R$ if the following are true:

$(x_{1}, x_{2}, \dots, x_{n}) \neq R$
$x_{i}$ is not a zero divisor in $R / (x_{1}, x_{2}, \dots, x_{i - 1})$

The notion generalizes to that of regular sequence on a module.

Revision as of 14:06, 5 May 2008 (view source) Vipul (talk \| contribs) (New page: ==Definition== Suppose <math>R</math> is a commutative unital ring, and <math>x_1, x_2, \ldots, x_n</math> is a sequence of elements in <math>R</math>. We say that the <math>x_i</math...)		Latest revision as of 16:34, 12 May 2008 (view source) Vipul (talk \| contribs) m (2 revisions)
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	* <math>x_i</math> is not a zero divisor in <math>R/(x_1,x_2,\ldots,x_{i-1})</math>		* <math>x_i</math> is not a zero divisor in <math>R/(x_1,x_2,\ldots,x_{i-1})</math>

	The notion generalizes to that of [[regular sequence of a module]].		The notion generalizes to that of [[regular sequence on a module]].