Math and stuff

Proposition Let $I(k) = \frac{1}{2\pi} \int_{0}^{2\pi} e^{ikt} dt$. Show that $I(0) = 1$. Show that $I(k) = 0$ if $k$ is a nonzero integer. What is $I(1/2)$? Solution 1 $I(0) = \frac{1}{2\pi}\int_{0}^{2\pi} 1dt = 1$. 2 \[\begin{align*} I(k) &= \frac{1}{2\pi} \int_{0}^{2\pi} e^{ikt} dt \\ &= \frac{1}{2\pi} \int_{0}^{2\pi} \cos(kt) + i\sin(kt)...

Proposition If $f: A \rightarrow B$ is a ring homomorphism and $M$ is a flat $A$-module, then $M_B = B \otimes_A M$ is a flat $B$-module. Lemma We will show in general that a ring $A$ is a flat $A$-module. Let $f: M \rightarrow M’$ be injective. Then $f \otimes...

Proposition Let $A, B$ be rings, let $M$ be an $A$-module, $P$ a $B$-module and $N$ an $(A, B)$-bimodule. Then $M \otimes_A N$ is naturally a $B$-module, $N \otimes_B P$ an $A$-module, and we have \[\begin{align*} (M \otimes_A N) \otimes_B P \simeq M \oplus (N \oplus_B P). \end{align*}\] Solution Part...

Proposition A ring $A$ is Boolean if $x^2 = x$ for all $x \in A$. In a Boolean ring $A$, show that $2x = 0$ for all $x \in A$; every prime ideal $p$ is maximal, and $A/p$ is a field with two elements; every finitely generated ideal in $A$...

Proposition Prove $\Ann(M + N) = \Ann(M) \cap \Ann(N)$. $(N:P) = \Ann((N + P) / N)$. Solution 1 $\Ann(M + N) = (0:M + N)$, and $\Ann(M) \cap \Ann(N) = (0:M) \cap (0:N)$. We showed that $(0:M + N) = (0:M) \cap (0:N)$ in a previous post. 2 \[\begin{align*} \Ann((N...