# The Azimuth Project λ-ring (Rev #1)

## Definition

A commutative ring $K$ is called a $\lambda$-ring if we are given a family of set operations $\lambda^k \colon K \to K$ for $k \geq 0$ such that for all $x,y \in K$

1. $\lambda^0(x) = 1$ and $\lambda^1(x) = x$
2. \$\lambda^k(x+y) = \sum_{i=0}^k \lambda^i(x) \lambda^{k-i}(y).

Note: there is a group homomorphism $\lambda_t$ from the additive group of $K$ to the multiplicative group $W(K) = 1+tK[[t]]$

A special $\lambda$-ring is a $\lambda$-ring $K$ such that the group homomorphism $\lambda_t \colon K \to W(K)$ is a $\lambda$-ring homomorphism.

## References

• Charles A. Weibel, The $K$-book: An introduction to algebraic $K$-theory, American Mathematical Society, Graduate Studies in Mathematics Volume 145, 2013.