TITLE: Rings in which every non-unit is a zero divisor
QUESTION [44 upvotes]: Is there a special name for the class of (commutative) rings in which every non-unit is a zero divisor? The main example is $\mathbf{Z}/(n)$. Are there other natural or interesting examples?

REPLY [6 votes]: How about calling these things balanced commutative rings?
Recall that a category satisfying "monic + epic --> isomorphism" is said to be balanced. Therefore:

a monoid is balanced iff every element that is both left-cancellative and right-cancellative is a unit.
a commutative monoid is balanced iff every element that is cancellative is a unit.

It makes sense to apply this terminology to rings, too:

a ring is balanced iff every element that is neither a left zero-divisor nor a right-zero divisor is a unit.
a commutative ring is balanced iff every non-zero divisor is a unit.