Least k Such that a^k=1 in Finite Ring R

[modnarandom]

Is there a term for the least k such that a^k = 1 in some ring R (if it exists, specifically in a finite ring)? Is there a way to get an upper bound in general? I know that it's hard to get even in say F_p, but I'm just looking for some conditions that would help me understand what it looks like.

 

[micromass]

Is there a term for the least k such that a^k = 1 in some ring R (if it exists, specifically in a finite ring)?

The term order should work well.

Is there a way to get an upper bound in general?

Let $n$ be the number of invertible elements in the ring, then $n$ is an upper bound.

I know that it's hard to get even in say F_p

Well, you know that the invertible elements in $\mathbb{F}_p$ form a cyclic group of order $p-1$. So finding the orders of the elements in equivalent to finding the orders of elements in $\mathbb{Z}_{p-1}$, which is easy. So you know exactly which orders show up and which don't. The hard part is finding an explicit isomorphism between $\mathbb{Z}_{p-1}$ and the invertible elements in $\mathbb{F}_p$ (or equivalently: finding an element in $\mathbb{F}_p$ with order $p-1$).