# Binomial ring

In mathematics, a binomial ring is a commutative ring whose additive group is torsion-free and contains all binomial coefficients

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

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 for which all Adams operations 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, 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), Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, ISBN 978-0-902480-06-3, MR 0283083
• Yau, Donald (2010), Lambda-rings, Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., ISBN 978-981-4299-09-1, MR 2649360