Is 1 always a modular inverse of 1?

Question

I have been looking for an answer online but I am unable to find one. Are inverses defined as to not include the number 1?

Would this mean that 1 is an inverse of itself in

$1*1^{-1}=1 (\textrm{mod p})$

where p is any number?

Answer 1

Yes: $1 \cdot 1 = 1 \equiv 1 \bmod m$ for all $m$.

Recall the definition: $b$ is an inverse of $a$ mod $m$ iff $ab \equiv 1 \bmod m$.