TITLE: Does there exist a system such that the additive identity is non-zero?
QUESTION [20 upvotes]: I am trying to explain how although the additive identity is written as $0$, it is not the same as the number $0$.
For example for a $2\times 2$ matrix the additive identity is $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$. However this is a bad example since it only involves $0$'s 
So what is a system such that the additve identity is non-zero (preferably not involving a $0$)?
I would use mod p but this (seemingly) contradicts the rule for the uniqueness of the identity
This arose because of a question, find a vector space such that $0=1$, however by "$0,1$" they meant the additive and multiplicative identities. So I was trying to explain that the additve identity is different from $0$ in certain systems.

REPLY [2 votes]: How about string addition?  The additive identity is an empty string.