Let $F =$ the free group generated by $\Sigma \cup V$ where $\Sigma = \Sigma(s)$ is the minimal alphabet that $s$ is over, and $V = \{V_1, \dots, V_n\}$ is a finite set of variables. I would like to quotient by the smallest normal subgroup containing all of $V_i^{-1} v_i$ where $v_i = $ a substr...

Here is another solution using only the Cayley-Hamilton Theorem for finitely generated modules (Proposition 2.4. in Atiyah-Macdonald) which, even though looks quite innocent, is a very powerful statement. Assume by contradiction that there is an injective map $\phi: A^m \to A^n$ with $m>n$. The ...

From a measure theory perspective how do I understand, notate, and evaluate the following, where $\pi(x)$ is the prime counting function. I think I need to choose a measure $\mu$ on a set $X.$ Also, what is the correct notation for the sum and integral below? $$ \sum_{1/100}^1 \pi(1/x)=~?$$ $$...