Question about a near $\Bbb{Z}$-semimodule that has trivial addition.

Question

https://en.wikipedia.org/wiki/Semimodule

Suppose that we have a structure $M$ that has:

1. Closure under addition.
2. Closure under multiplication by any $a \in\Bbb{Z}$.
3. The addition of any two elements is constant: $x + y = c \in M$ for some fixed constant.
4. (Doesn't necessarily have) a neutral element. If it does have a neutral element then (3) doesn't apply to adding $0$ i.e. $0 + x = x = c$ is not necessary unless $x = c$.

Then can we conclude that $M$ is isomorphic to the trivial module?

Answer 1

With the current rules, $\mathbb Z/2\mathbb Z$ is a nontrivial example: since $0$ is excluded from rule 3, the only sum that rule 3 applies to is $1+1 = 0$.

Generalizing this, we can have $M = \{0, x_1, x_2, \dots, x_n\}$ with $0 + x_i = x_i + 0 = x_i$ and $x_i + x_j = 0$ for all $i,j$. Let $m0 = 0$ for all $m \in \mathbb Z$. Let $m x_i = 0$ for even $m$ and $mx_i = x_i$ for odd $m$.

If there is no neutral element (and so rule 3 applies to any sum of elements), then for any $x\in M$, $x = 1x = (0+1)x = 0x+1x = c$, so $M$ is trivial.