Math and stuff

Proposition Let $p$ be a prime ideal in $A$. Show that $p[x]$ is a prime ideal in $A[x]$. If $m$ is a maximal ideal in $A$, is $m[x]$ a maximal ideal in $A[x]$? Solution Let $a(x) = a_nx^n + \cdots + a_0, b(x) = b_mx^m + \cdots + b_0 \in...

Proposition Find the image of the following points under the stereographic projection $\phi$: $(0, 0, -1)$, $(0, 0, 1)$, $(1, 0, 0)$, $(0, 1, 0)$, $(1, 1, 0)$. Solution $(0, 0, -1) \mapsto (0, 0, 0)$, $(0, 0, 1) \mapsto \infty$, $(1, 0, 0) \mapsto (1, 0, 0)$, $(0, 1,...

Proposition Find a Mobius transformation that maps the unit circle to $\{ x + iy \in \mathbb{C} : x + y = 0 \}$. Find two Mobius transformations that map the unit disk $\{ z \in \mathbb{C} : \abs{z} < 1 \}$ to $\{ x + iy \in \mathbb{C} :...

Proposition Let $A$ be a ring and let $A[x]$ be the ring of polynomials in an indeterminate $x$, with coefficients in $A$. Let $f = a_0 + a_1x + \cdots + a_nx^n \in A[x]$. Prove that $f$ is a unit in $A[x] \iff a_0$ is a unit in $A$ and...

Proposition In the ring $A[x]$, the Jacobson radical is equal to the nilradical. Solution Since every maximal ideal is prime, $N(A[x]) \subset J(A[x])$. Let $f \in J(A[x])$. By Proposition 1.9[Atiyah], $1 - xf$ is a unit in $A[x]$. As we showed previously, this implies $a_0, \cdots, a_n \in A$ are...