# Necessary and sufficient conditions for a ring to have no zero divisors.

Pehaps I missed something, but in my first abstract algebra course we were taught that if in a ring either cancellation law (i.e. $ac = bc, c\neq 0 \Rightarrow a=b$; either this one or the left one), then said ring had no zero divisors. The converse holds too: if a ring has no zero divisors, both cancellation laws hold.

My first question is: Does this mean that any division ring has no zero divisors?; because, in a division ring, if one has $ac = bc$ with $c\neq 0$, one could always multiply both sides by $c^{-1}$ and $ac = bc$ would imply that $a=b$ and, thus, the right cancellation law would hold.

My second question is: Are there more necessary and sufficient conditions for a ring to have no zero divisors?

• Presumably you mean no proper zero divisors for your first question, in which case the answer is yes. – Batman Mar 30 '15 at 3:08
• I'm sorry, what do you mean with "proper"?, elements different from $0$? – Miguelgondu Mar 30 '15 at 3:08

## 1 Answer

Not sure if you are aware of this, but the rings you are referring to are called integral domains. As for more necessary/sufficient conditions, this might help:

http://en.wikipedia.org/wiki/Integral_domain#Definitions

• I knew them by name, but Wikipedia had just what I was looking for and I hadn't noticed. Thanks. – Miguelgondu Mar 30 '15 at 3:55