Let \(A\) be a commutative ring, and let \(S\) be a multiplicative closed subset of \(A\text{.}\) Give an example where \(A\to S^{-1}A\) given by \(a\mapsto \tfrac{a}{1}\) is not injective.
2.
Let \(A\) be an integral domain, and let \(S\) be a multiplicative closed subset of \(A\) such that \(0\not\in S\text{.}\) Show that \(S^{-1}A\) is an integral domain. Is the converse true?
3.
Recall that a field is said to be prime if it is isomorphic to \(\Q\) or one of the finite fields \(\mathbb{F}_p\text{.}\) Let \(P\) be a prime field and \(R\) be a ring. If \(f,g\) are ring homomorphisms from \(P\to R\) then show that \(f=g\text{.}\)
4.
A field is said to be perfect if its characteristic is zero or if its characteristic is \(p\neq 0\) then the mapping \(x\mapsto x^p\) is surjective. Show the following.
The field \(\mathbb{F}_p\) is perfect.
Every prime field is perfect.
The field of rational functions over \(\mathbb{F}_p\text{,}\)
\begin{equation*}
\mathbb{F}_p(X)=\big\{u(X)/v(X):u(X),v(X)\in\mathbb{F}_p[X] \text{ and }v(X)\neq 0\big\}
\end{equation*}
