Math and stuff

Proposition Let $A$ be a ring and let $X$ be the set of all prime ideals of $A$. For each subset $E$ of $A$, let $V(E)$ denote the set of all prime ideals of $A$ which contain $E$. Prove that if $a$ is the ideal generated by $E$, then $V(E)...

Proposition Let $A$ be a ring, $\mathfrak{R}$ its nilradical. Show that the following are equivalent: $A$ has exactly one prime ideal; every element of $A$ is either a unit or nilpotent; $A/\mathfrak{R}$ is a field. Solution By Proposition 1.8[Atiyah], $\mathfrak{R}$ is the intersection of all the prime ideals of $A$....

Proposition Evaluate the integrals $\int_{\gamma} xdz, \int_{\gamma} ydz, \int_{\gamma}zdz$ and $\int_{\gamma}\overline{z}dz$ along each of the following paths. $\gamma$ is the line segment from $0$ to $1 - i$. $\gamma = C[0, 1]$. $\gamma = C[a, r]$ for some $a \in \mathbb{C}$. Solution 1 $\gamma(t) = t - it$ with $0...

Proposition Prove Proposition 4.2 (A first course in complex analysis) and the fact that the length of $\gamma$ does not change under reparametrization. Solution Let $\tau:[c, d] \rightarrow [a, b]$ be given such that $\sigma = \gamma \circ \tau$. \[\begin{align*} \int_{c}^{d} f(\sigma(t))\sigma'(t) dt &= \int_{c}^{d} f((\gamma \circ \tau)(t))(\gamma \circ \tau)'(t)...

Proposition If $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $g: \mathbb{R}^m \rightarrow \mathbb{R}^p$, show that \((g \circ f)_{\ast} = g_{\ast} \circ f_{\ast}\) and $(g \circ f)^{\ast} = f^{\ast} \circ g^{\ast}$. If $f, g: \mathbb{R}^n \rightarrow \mathbb{R}$, show that $d(f \cdot g) = f \cdot dg + g \cdot df$. Solution 1 Let...