# The Unapologetic Mathematician

## 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 $1=\bar{1}$.