Is $( \mathbb{ Z}_{10}^{*},\cdot) \rightarrow (\mathbb{Z}_5^{*},\cdot), n\pmod {10} \mapsto n \pmod 5 $ well-defined?
So what I think is that it is not because the odd multiples of 5 in $\mathbb{ Z}_{10}^{*}$ map to the class of $5\equiv0$ which is the only class taken out of $\mathbb{Z}_5$ to yield $\mathbb{Z}_5^{*}$, and since not all elements in the domain are being mapped, then it is not a mapping, ie it is not well-defined(in a mapping all elements of the first set must be part of the domain)
What do you think? Feel free to elaborate
The first question you need to ask to determine whether the map is well defined is whether the following always holds for $x, y \in \Bbb Z_{10}^*$:
$$x \equiv y \pmod{10} \Rightarrow x \equiv y \pmod{5}.$$
And that's obviously true: $x \equiv y \pmod{10} \Rightarrow 10 \mid y-x \Rightarrow 5 \mid y-x \Rightarrow x \equiv y \pmod{5}$.
And it's also clearly true that $x \in Z_{10}^* \iff (x, 10)=1 \Rightarrow (x, 5)=1 \iff x \in Z_5^*.$
The map is therefore well defined.
Now whether the map is a homomorphism is an entirely separate question.