Suppose I have two algebraic objects $\mathcal{A}, \mathcal{B}$ , and I want to define a map between $\mathcal{A}, \mathcal{B}$. So what is a well-defined map and why it has to be well-defined(may be it's a stupid question), and what will happened if it is not well-defined? I read about this in Tim Gowers blog, I got the point however I could not make it clear in algebraic case.
I also have another question. Suppose that $R$ and $S$ are two rings, $\varphi : R \longrightarrow S$ is a ring homomorphism, $\mathfrak{p} \in $Spec$R$, let $I$ be the ideal generated by $\varphi(\mathfrak{p})$ in $S$, $U:=\lbrace\varphi(r)+I|r\in R-\mathfrak{p}\rbrace $. Then we can form the ring $S_{[\mathfrak{p}]}:=U^{-1}(S/I)$. Can we defined a map from $S$ to $S_{[\mathfrak{p}]}$ through two maps $\pi : S\longrightarrow S/I$ and $f:S/I \longrightarrow U^{-1}(S/I)$? I think we can make a map from $S\times (R-\mathfrak{p})$ to $S_{[\mathfrak{p}]}$. From this, I have to think again about the well-defined property of it, though I still do not know what exactly it means.
I know that my question is not well-written, since my idea in my head still complicated, I beg your pardon for this. Thanks for reading and please feel freely commenting and answering my question.