Verification on explanation of how one ring has unity and the other does not One of the examples my instructor gave us is that $(\mathbb{Z},+,\ast)$ has unity while $(n\mathbb{Z}, +,\ast)$ does not.  Is the reason that $\mathbb{Z}$ has unity that $1$ is guaranteed to be in $\mathbb{Z}$ but not guaranteed in $n\mathbb{Z}$?  

Almost! $\mathbb{Z}$ is $\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$ which has $1$ as the multiplicative identity $\mathbf{1}$ (or unity, as your professor calls it), but it is important to note that the unity of a ring may not always be the number $1$ (consider the ring of $2\times2$ matrices for example). 
However, for the purposes of this question, we find that in the set $2\mathbb{Z} = \{\ldots -6, -4, -2, 0, 2, 4, 6 \ldots\}$ there does not exist an identity element $\mathbf{1}$ such that for every $x \in 2\mathbb{Z}$, $\mathbf{1}x = x = \mathbf{1}x$. So $n\mathbb{Z}$ may not have a multiplicative identity/unity. 
