Math and stuff

Proposition Let $M$ be an $A$-module. If every non-empty set of finitely generated submodules of $M$ has a maximal element, then $M$ is Noetherian. Solution By Proposition 6.2[Atiyah], it suffices to show that every submodule of $M$ is finitely generated. Let $N$ be a submodule of $M$. If $N =...

Proposition Let $A \subset B$ be rings, $B$ integral over $A$. If $x \in A$ is a unit in $B$, then it is a unit in $A$. The Jacobson radical of $A$ is the contraction of the Jacobson radical of $B$. Solution 1 Let $x \in A$ and suppose $x$...

Proposition Let $f: B \rightarrow B’$ be a homomorphism of $A$-algebras, and let $C$ be an $A$-algebra. If $f$ is integral, prove that $f \otimes 1: B \otimes_A C \rightarrow B’ \otimes_A C$ is integral. Solution Each element in $B’ \otimes_A C$ is a finite sum $\sum_{i=1}^{n} b_i’ \otimes c_i$...

Proposition If $M$ is a $k$-dimensional manifold-with-boundary, prove that $\partial M$ is a $(k - 1)$-dimensional manifold and $M \setminus \partial M$ is a $k$-dimensional manifold. Solution Let $p \in \partial M$. Then there exists an open subset $U$ containing $p$ and an open subset $V \subset \mathbb{R}^n$ with a...

Proposition If $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$, the graph of $f$ is $\{ (x, y) \mid y = f(x) \}$. Show that the graph of $f$ is an $n$-dimensional manifold if and only if $f$ is differentiable. Solution Suppose $f$ is differentiable. Let $g: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^m$ be defined...