Shamir's ID based signatures - Notation issues
Ok, lets look at the operations.

Sign: $s = g * r^{f(t,m)} \pmod n$

This is an assignment. You compute $(g * r^{f(t,m)}) \mod n$ and assign the resulting value to $s$. If you have a multiplication $(a \cdot b) \mod n$, this is equal to $((a \mod n)\cdot (b \mod n)) \mod n$. See for instance here.

Verification: $s^e = i * t^{f(t, m)} \pmod n$

This is no check for equality, but a check for congruence. 

You can implement this by computing $s^e \mod n$ as well as $(i * t^{f(t, m)}) \mod n$ and check if these values are equal. 

fgrieu has recently written an answer to a recent question on congruences which clarifies this nicely.