Math and stuff

Proposition Use induction on $n$ to show that $[a_1, b_1] \times \cdots \times [a_n, b_n]$ is not a set of measure 0 (or content 0) if $a_1 < b_i$ for each $i$. Solution By Theorem 3-6[Spivak], it suffices to prove this for measure 0. By Theorem 3-5[Spivak], the statement is...

Proposition If $C$ is a bounded set of measure 0 and $\int_A \chi_C$ exists, show that $\int_A \chi_C = 0$. Solution Let $P$ be a partition. Suppose $L(f, P) \ne 0$. Since $\chi_C(x)$ is either 0 or 1 for any $x \in A$, $m_S$ is 0 or 1 for any...

Proposition Prove that $[a_1, b_1] \times \cdots \times [a_n, b_n]$ does not have content 0 if $a_i < b_i$ for each $i$. Solution We will use the same approach that the proof of Theorem 3-5 uses. Let $A = [a_1, b_1] \times \cdots \times [a_n, b_n]$. Let $\{ U_1, \cdots,...

Proposition If $M$ and $N$ are flat $A$-modules, then so is $M \otimes_A N$. If $B$ is a flat $A$-algebra and $N$ is a flat $B$-module, then $N$ is flat as an $A$-module. Solution 1 Let $f: L’ \rightarrow L$ be an injective $A$-module homomorphism. Since $M$ is flat, $f...

Chapter 1: Rings and Ideals Problem 1 Problem 2 Problem 4 Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 Problem 11 Problem 12 Problem 13 Problem 14 Problem 15 Problem 16 Problem 17 Problem 18(in-text) Problem 18 Problem 19 Problem 22 Chapter 2: Modules Problem 1 Problem 2(in-text)...