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I am not sure if I wrote this wrong or I am just not getting it.

In some of my notes, I have written $(\mathbb{Z}^*, .)$ is a Monoid and not a group (no inverse).

The asterisk means the set of integers without 0 and the . is multiplication.

I don't think this is correct. Isn't because we took away 0 that we have inverses under multiplication?

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    Ohh the rationals isn't an integers anyways. Thank you. NOW it makes sense to me2012-10-10

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Since the identity element is $1$, it's obvious that the inverse of $n\in \Bbb Z$ is not in $\Bbb Z$, e.g. $\frac12 \notin \Bbb Z$. Excluding $n=1$, which is self-inverse.

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    Also excluding $-1$ which is also self-inverse.2012-10-10