Math and stuff

Proposition A ring $A$ is such that every ideal not contained in the nilradical contains a nonzero idempotent (that is, an element $e$ such that $e^2 = e \ne 0$). Prove that the nilradical and Jacobson radical of $A$ are equal. Solution When $A = (0)$, this is trivial. Suppose...

Proposition A local ring contains no idempotent $\ne 0, 1$. Solution Let $A$ be a local ring that contains no idempotent $\ne 0, 1$. Let $x$ be an idempotent $\ne 0, 1$. If no such element exists, we are done. Suppose it is a unit and let $y$ be the...

Proposition Let $a$ be an ideal $\ne (1)$ in a ring $A$. Show that $a = r(a) \iff a$ is an intersection of prime ideals. Solution Suppose $a = r(a)$. By Proposition 1.14[Atiyah], $r(a)$ is the intersection of the prime ideals containing $a$. Therefore, $a$ is an intersection of prime...

Proposition Integrate the function $f(z) = \overline{z}$ over the following paths: $\gamma(t) = t + it (0 \leq t \leq 1)$. $\gamma(t) = t + it^2 (0 \leq t \leq 1)$. $\gamma_1(t) = t$ with $0 \leq t \leq 1$, and $\gamma_2(t) = 1 + it$ with $0 \leq t...

Proposition Integrate the following functions over the circle $C[0, 2]$: $f(z) = z + \overline{z}$. $f(z) = \frac{1}{z^4}$. $f(z) = z^2 - 2z + 3$. $f(z) = xy$. Solution The circle can be parametrized as $\gamma(t) = 2\exp(it) = 2\cos(t) + 2i\sin(t)$. Then $\gamma’(t) = 2i\exp(it) = -2\sin(t) + 2i\cos(t)$....