# 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.

• 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 $1=\bar{1}$.