TITLE: If every element of R has a (multiplicative) inverse, then $R = \{0\}$
QUESTION [7 upvotes]: I am struggling to understand why is this the case. I need to prove this, but I don't understand how it's true. For example, if every non-zero element of $R$ has a multiplicative inverse, then it's a field.  So how does $R=\{0\}$?
Thanks you for your time :)

REPLY [3 votes]: First of all if every non zero element of $R$ has a multiplicative inverse that makes it a division ring and not a field a good example is the quaternions it only becomes a field if the multiplication is commutative. Secondly from the axioms of ring theory $0$ does not have a multiplicative inverse except for the ring {$0$}