# The Unapologetic Mathematician

## R.I.P., Dr. Cohen

From Alexandre Borovik I hear that Paul Cohen passed yesterday. He was probably best known for showing that the continuum hypothesis and the axiom of choice are independent of Zermelo-Fraenkel set theory. If I find an actual news article about it I’ll update here.

## Ring exercises

As spring break comes to an end, it’s another travel day. As I head back to New Haven, I think I’ll leave a few basic theorems about rings that can be shown pretty much straight from the definitions. The first three hold in any ring, while the last two require the ring to have a unit (multiplicative identity).

• For any element $a$, $0a=a0=0$.
• For any elements $a$ and $b$, $(-a)b=a(-b)=-(ab)$. Remember that $-a$ is the inverse of $a$ in the underlying abelian group of the ring.
• For any elements $a$ and $b$, $(-a)(-b)=ab$.
• For any invertible elements $a$ and $b$, $(ab)^{-1}=b^{-1}a^{-1}$.
• The multiplicative identity is unique. That is, if there is another element $\bar{1}$ so that $\bar{1}a=a\bar{1}=a$ for all $a$, then $1=\bar{1}$.