Q:

performing a power operation ($a^n$) in a ring

In a ring - when performing a power operation, i.e $a^n$, to which operation is it related to? $+$ or $*$?
On one hand - I know that a power is defined on multiplication - in "regular" numbers, but it occurred to me that in these types of structures, it can be different.

A:

It is justified to ask this, but in the presence of multiplicative and additive notation, $a^n$ (with $n\in\mathbb N$) stands for $\underbrace{a \cdot a\cdot \ldots \cdot a}_n$ and (even if $n$ is an element of $\mathbb Z$ and not of the ring) $n\cdot a$ stands for $\underbrace{a+a+\ldots +a}_n$.
Note that already for abelian groups it is usual to write $n\cdot a$ when the group operation is written additively. The notation $a^n$ is used for "anything but $+$" (so for $\cdot$, $*$, $\circ$  and so on as operational symbols).

