Math and stuff

Proposition Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be given by $f(z) = z^2 - 2$. Find the maximum and minimum of $\abs{f(z)}$ on the closed unit disk. Solution $\abs{z^2 - 2} \leq \abs{z^2} + \abs{-2} \leq 3$. $\abs{f(i)} = 3$, so 3 is the maximum value. $\abs{2} \leq \abs{2 - z^2}...

Proposition Prove a partial converse of Theorem 5-1: If $M \subset \mathbb{R}^n$ is a $k$-dimensional manifold and $x \in M$, then there is an open set $A \subset \mathbb{R}^n$ containing $x$ and a differentiable function $g: A \rightarrow \mathbb{R}^{n - k}$ such that $A \cap M = g^{-1}(0)$ and $g’(y)$...

Proposition Let $a$ be an irreducible ideal in a ring $A$. Then the following are equivalent: $a$ is primary; for every multiplicatively closed subset $S$ of $A$ we have $(S^{-1}a)^{c} = (a:x)$ for some $x \in S$; the sequence $(a:x^n)$ is stationary, for every $x \in A$. Solution $1 \implies...

Proposition Let $M$ be a Noetherian $A$-module and $u: M \rightarrow M$ a module homomorphism. If $u$ is surjective, then $u$ is an isomorphism. If $M$ is Artinian and $u$ is injective, then again $u$ is an isomorphism. Solution 1 $\ker(u) \subset \ker(u^1) \subset \ker(u^2) \subset \cdots$ is an ascending...

Proposition Let $A$ be a subring of a ring $B$, such that the set $B - A$ is closed under multiplication. Show that $A$ is integrally closed in $B$. Solution Let $C$ be the integral closure of $A$ in $B$. Clearly, $A \subset C$. Suppose $A \subsetneq C$. Let $x...