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A ring is a zero ring if the product of any two elements is the additive identity (or zero).
Zero rings are commutative under multiplication. For if  is a zero ring,  is its additive identity, and  , then
Every zero ring is a nilpotent ring. For if  is a zero ring, then
 .
Since every subring of a ring must contain its zero element, every subring of a ring is an ideal, and a zero ring has no prime ideals.
The simplest zero ring is
 . Up to isomorphism, this is the only zero ring that has a multiplicative identity.
Zero rings exist in abundance. They can be constructed from any ring. If  is a ring, then
considered as a subring of
 (with standard matrix addition and multiplication) is a zero ring. Moreover, the cardinality of this subset of
 is the same as that of  .
Moreover, zero rings can be constructed from any abelian group. If  is a group with identity  , it can be made into a zero ring by declaring its addition to be its group operation and defining its multiplication by
 for any  .
Every finite zero ring can be written as a direct product of cyclic rings, which must also be zero rings themselves. This follows from the fundamental theorem of finite abelian groups. Thus, if
 are distinct primes,
 are positive integers, and
 , then the number of zero rings of order  is
 , where  denotes the partition function.
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