Regular Sequence

Regular Sequence

In commutative algebra, if R is a commutative ring and M an R-module, a nonzero element r in R is called M-regular if r is not a zerodivisor on M, and M/rM is nonzero. An R-regular sequence on M is a d-tuple

r1, ..., rd in R

such that for each i ≤ d, ri is Mi-1-regular, where Mi-1 is the quotient R-module

M/(r1, ..., ri-1)M.

Such a sequence is also called an M-sequence.

An R-regular sequence is usually called simply a regular sequence.

It may be that r1, ..., rd is an M-sequence, and yet some permutation of the sequence is not. It is, however, a theorem that if R is a local ring or if R is a graded ring and the ri are all homogeneous, then a sequence is an R-sequence only if every permutation of it is an R-sequence.

The depth of R is defined as the maximum length of a regular R-sequence on R. More generally, the depth of an R-module M is the maximum length of an M-regular sequence on M. The concept is inherently module-theoretic and so there is no harm in approaching it from this point of view.

The depth of a module is always at least 0 and no greater than the Krull dimension of the module.

Read more about Regular Sequence:  Examples

Famous quotes containing the words regular and/or sequence:

    “A regular council was held with the Indians, who had come in on their ponies, and speeches were made on both sides through an interpreter, quite in the described mode,—the Indians, as usual, having the advantage in point of truth and earnestness, and therefore of eloquence. The most prominent chief was named Little Crow. They were quite dissatisfied with the white man’s treatment of them, and probably have reason to be so.”
    —Henry David Thoreau (1817–1862)

    “We have defined a story as a narrative of events arranged in their time-sequence. A plot is also a narrative of events, the emphasis falling on causality. “The king died and then the queen died” is a story. “The king died, and then the queen died of grief” is a plot. The time sequence is preserved, but the sense of causality overshadows it.”
    —E.M. (Edward Morgan)