# Binomial ring

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In mathematics, a **binomial ring** is a [commutative ring](/source/Commutative_ring) whose additive [group](/source/Group_(mathematics)) is [torsion-free](/source/Torsion_(algebra)) and contains all [binomial coefficients](/source/Binomial_coefficient)

- \binom{x}{n} = \frac{x(x-1)\cdots(x-n+1)}{n!}

for *x* in the ring and *n* a positive integer. Binomial rings were introduced by Hall (1969).

Elliott (2006) showed that binomial rings are essentially the same as [λ-rings](/source/Lambda_ring) for which all [Adams operations](/source/Adams_operation) are the identity.

## References

- Elliott, Jesse (2006), "Binomial rings, integer-valued polynomials, and λ-rings", *Journal of Pure and Applied Algebra*. **207** (1): 165–185, [doi:10.1016/j.jpaa.2005.09.003](https://doi.org/10.1016/j.jpaa.2005.09.003). [ISSN 0022-4049](https://www.worldcat.org/issn/0022-4049). MR 2244389
- Hall, Philip (1969 [1957]), [*The Edmonton notes on nilpotent groups. Notes of lectures given at the Canadian Mathematical Congress Summer Seminar (University of Alberta, 12–30 august 1957)*](https://books.google.com/books?id=eeruAAAAMAAJ), Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, ISBN 978-0-902480-06-3. MR 0283083
- Yau, Donald (2010), [*Lambda-rings*](https://books.google.com/books?id=d7vKnjxyvxQC), Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., ISBN 978-981-4299-09-1. MR 2649360

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