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:

    The solid and well-defined fir-tops, like sharp and regular spearheads, black against the sky, gave a peculiar, dark, and sombre look to the forest.
    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)