Let \(F\) be a field and let \(M_n(F)\) be the ring of \(n\times n\) matrices over \(F\text{.}\) Show \(\mathcal{Z}(M_n(F))\simeq F\text{.}\)
2.
Describe all ring homomorphism from \(\Z/n\Z\) to \(\Z/m\Z\text{.}\)
3.
Find \(\Aut_\Rings(\Z)\) and \(\Aut_\Rings(\Z/n\Z)\text{.}\)
4.
Show that the following rings are not isomorphic.
\(\Z/n\Z\not\simeq\Z/m\Z\) for \(m\neq n\)
\(\displaystyle \Z[X]\not\simeq\Q[X]\)
5.
Does there exists a ring homomorphism from \(\Q\) to \(\Z\text{?}\)
6.
Let \(F\) be a field, and let \(a_1,\ldots,a_n\in F\text{.}\) Show that \(\ev_{(a_1,\ldots,a_n)}\colon F[X_1,\ldots,X_n]\to F\) is an \(F\)-algebra homomorphism.
