Q:

Understanding the notation $a \equiv b \bmod p \mathbb{Z} [\zeta]$

Im trying to follow Keith Conrad's notes on Fermats Last Theorem for regular primes but I'm having trouble with some unfamiliar notation. Half way down page three he begins talking about a congruence which I'm not sure about. Namely, $y(\zeta - \zeta^{-1})$$  \equiv 0 $ mod p$\mathbb Z[\zeta]$. 
My initial thought was that perhaps it's just reminding me what ring we are working in and I just just treat it as (mod p) but  he uses (mod p) many times so there must be some subtle difference I'm not grasping. (He also uses mod p$\mathbb Z$)

A:

The notation is $a\equiv b\bmod I$ for elements $a,b$ of a ring $R$ and an ideal $I$ of $R$. It means that $b-a\in I$. For $R=\mathbb{Z}$ we would have $I=n\mathbb{Z}$ for some integer $n$, and then we obtain the usual congruence.
In the text on Fermat the ring is $R=\mathbb{Z}[\zeta]$, the ring of integers in the cyclotomic field $\mathbb{Q}[\zeta]$, with the ideal $I=(p)=p\mathbb{Z}[\zeta]$.

