# Binomial ring

2010 Mathematics Subject Classification: Primary: 13F [MSN][ZBL]

A ring $R$ with torsion-free addition in which all binomial symbols $$\binom{x}{n} = \frac{x(x-1)\cdots(x-n+1)}{n!}$$ are well-defined as functions on $R$: the corresponding elements of $R \otimes_{\mathbf{Z}} \mathbf{Q}$ lie in $R$. Clearly any field of characteristic zero is a binomial ring, as is the ring of integers $\mathbf{Z}$.

If $R$ is binomial, then defining $\lambda^n (a) = \binom{a}{n}$ makes $R$ a lambda-ring with operators $\lambda^n$. The Adams operations are all equal to the identity.

#### References

• Yau, Donald Lambda-rings World Scientific (2010) ISBN 978-981-4299-09-1 Zbl 1198.13003
How to Cite This Entry:
Binomial ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Binomial_ring&oldid=39491