Math and stuff

Proposition Let $A$ be a ring $\ne 0$ and let $\Sigma$ be the set of all multiplicatively closed subsets $S$ of $A$ such that $0 \notin S$. Show that $\Sigma$ has maximal elements, and that $S \in \Sigma$ is maximal if and only if $A \setminus S$ is a minimal...

Proposition Let $A$ be a ring. Suppose that, for each prime ideal $p$, the local ring $A_p$ has no nilpotent element $\ne 0$. Show that $A$ has no nilpotent element $\ne 0$. If each $A_p$ is an integral domain, is $A$ necessarily an integral domain? Solution $N(A)$, the nilradical of...

Proposition Suppose $f$ has a simple pole at $z_0$ and $g$ is holomorphic at $z_0$. Prove that \[\begin{align*} \Res_{z = z_0}(f(z)g(z)) = g(z_0)\Res_{z = z_0}(f(z)). \end{align*}\] Solution By Corollary 9.6[A first course in complex analysis], $f(z) = \tilde{f}(z)/(z - z_0)$ for some $\tilde{f}$ that is holomorphic in an open disk...

Proposition Use the Weierstrass M-test to show that each of the following series converges uniformly on the given domain. $\sum_{k \geq 1} \frac{z^k}{k^2}$ on $\overline{D}[0, 1]$. $\sum_{k \geq 0} \frac{1}{z^k}$ on $\{ z \in \mathbb{C} : \abs{z} \geq 2 \}$. $\sum_{k \geq 0} \frac{z^k}{z^k + 1}$ on $\overline{D}[0, r]$ where...

Proposition Find a function representing each of the following series. $\sum_{k \geq 0} \frac{z^{2k}}{k!}$ $\sum_{k \geq 1} k(z - 1)^{k - 1}$ $\sum_{k \geq 2} k(k - 1)z^k$ Solution 1 Based on the Taylor series for $e^z$, it is easy to see that this series represents $e^{z^2}$. 2 $\frac{1}{1 -...