What is the definition of a ring element raised to the zeroth power? I have a fairly straightforward question about defining exponentiation in rings: Given any element $a \in R$ of any ring $R$, what is $a^0$ defined as? Is it the additive identity or the multiplicative identity (if it exists)?
In the context of groups we define $a^0 = e$ for whatever the identity of that one operation is. But in the context of rings, $a^n$ would refer to the ring product, so I would assume $a^0$ to be the multiplicative identity, but this doesn't always exist.
Is there any convention, or is it simply left undefined/context dependent?
Edit: In regards to being a duplicate of the $0^0$ question, I had already read through both the question and some answers, but I thought this was a 'fairly separate' question. In particular, this is about rings in general, and while there is a relationship, most of the discussion in the other question pertains explicitly to $\mathbb{R}$. For example discussion of non-integer powers, and limits as a motivating value for the definition, and the fact $\mathbb{R}$ has unity.
 A: Loosely speaking, for $a\in R$, $a^n$ means the product $a\cdots a$ consisting of $n$ factors. Rigorously, we could establish a right $\mathbb{N}^*$-pseudomodule over $(R,\cdot)$ whose module operation is $^\wedge:R\times \mathbb{N}^*\to R$. Here, a pseudomodule is a generalization of a semimodule that does not require the monoid to be commutative.
The properties of $R_{\mathbb{N}^*}$ are then

*

*$a^{n+m} = a^n\cdot a^m$

*$a^1 = a$
When $R$ is unitary, then we can use the right $\mathbb{N}$-pseudomodule which includes the property that
$$a^0=1_R$$
for $a\neq 0_R$. Thus in a unitary ring, a zeorth power is indeed the multiplicative identity of the ring.
If $R$ is a division ring, then we upgrade to a right $\mathbb{Z}$-pseudomodule and enable negative exponents. If $R$ is commutative then we can go to a $\mathbb{N}^*$-semimodule. If $R$ is a field, then we have the familiar $\mathbb{Z}$-semimodule of the rational numbers and the real numbers. Rational and then irrational exponents require further properties.
