I was recently given the following two-part problem on a homework assignment. To be clear from the start, I am not asking for a solution to the problem (I like to solve my own homework problems!) - but I believe that part (2) is incorrectly stated. Particularly I think the hypothesis needs to be stronger to get the desired result. My Question is: is my hypothesis true that, as stated, it is not enough to reach the desired claim.
The question reads:
a) Prove that if $\phi \colon R \to S$ is a ring homomorphism and $\phi(f)$ is a unit in $S$ then there is a unique ring homomorphism $\psi \colon R[t]/ \langle tf-1 \rangle \to S$ such that $\psi \circ i = \phi$ where $i$ is the natural embedding of $R$ into $R[t]/ \langle tf-1 \rangle$.
b)Prove that if $Q$ is any ring with the above property then $Q$ is isomorphic to $R[t]/ \langle tf-1 \rangle$.
As many of you will likely recognize, yes, this is leading towards proving the "Rabinowitz Trick".
I have successfully finished part a) and I know how the argument $\textit{should}$ go for part b) only because I am familiar with localization and the ring $R[t]/\langle tf-1 \rangle$ is just the localization $R_f$ in disguise, and I am familiar with the universal property that characterizes localizations, but as stated I don't think we have enough to prove the isomorphism. Perhaps because 'if $Q$ has the above property' is somewhat vague, but I already asked my professor once and he seemed confident I should be successful in applying the standard argument where I have compositions giving $Q \to Q$ and $R_f \to R_f$ both being the identity thus granting a unique isomorphism between them.
I am only posting here because I don't want to bother emailing my professor again unless I am sure there is a flaw in the problem statement. But I think we need to add the clause that the ring $Q$ comes with a morphism, say $j \colon R \to Q$ such that $j(f)$ is also a unit in $Q$