# System of parameters

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Mathematical concept in dimension theory of local rings

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In [mathematics](/source/Mathematics), a **system of parameters** for a [local](/source/Local_ring) [Noetherian ring](/source/Noetherian_ring) of [Krull dimension](/source/Krull_dimension) *d* with [maximal ideal](/source/Maximal_ideal) *m* is a set of elements *x*1, ..., *x**d* that satisfies any of the following equivalent conditions:

1. *m* is a [minimal prime](/source/Minimal_prime_ideal) over (*x*1, ..., *x**d*).

1. The [radical](/source/Radical_of_an_ideal) of (*x*1, ..., *x**d*) is *m*.

1. Some power of *m* is contained in (*x*1, ..., *x**d*).

1. (*x*1, ..., *x**d*) is [*m*-primary](/source/Primary_ideal).

1. R/(*x*1, ..., *x**d*) is an [Artinian ring](/source/Artinian_ring).

Every local Noetherian ring admits a system of parameters.[1]

It is not possible for fewer than *d* elements to generate an ideal whose radical is *m* because then the dimension of *R* would be less than *d*.

If *M* is a *k*-dimensional module over a local ring, then *x*1, ..., *x**k* is a **system of parameters** for *M* if the [length](/source/Length_of_a_module) of *M* / (*x*1, ..., *x**k*) *M* is finite.

## General references

- [Atiyah, Michael Francis](/source/Michael_Atiyah); [Macdonald, I. G.](/source/Ian_G._Macdonald) (1969), *[Introduction to Commutative Algebra](/source/Introduction_to_Commutative_Algebra)*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., [MR](/source/MR_(identifier)) [0242802](https://mathscinet.ams.org/mathscinet-getitem?mr=0242802)

- [Eisenbud, David](/source/David_Eisenbud) (1995), *Commutative Algebra*, Springer-Verlag, New York, [MR](/source/MR_(identifier)) [1322960](https://mathscinet.ams.org/mathscinet-getitem?mr=1322960)

## References

1. **[^](#cite_ref-Hochster_1-0)** ["Math 711: Lecture of September 5, 2007"](https://web.archive.org/web/20241231053848/https://dept.math.lsa.umich.edu/~hochster/711F07/L09.05.pdf) (PDF). University of Michigan. September 5, 2007. Retrieved March 9, 2026.

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