In particular, the zero ring is not a field.

This implies it can be localized only to a zero ring.

The trivial ring is an example of a zero ring.

Any subgroup of the additive group of a zero ring is an ideal.

Therefore, the concept of a zero ring is interesting only for rings that are not unital rings.

Any abelian group can be turned into a zero ring by defining the product of any two elements to be 0.

These authors require their rings to have unity, hence all zero rings are trivial.)

In the games, Sonic will die if he is attacked with zero rings, but with more rings, he can survive.

A functor from the category of abelian groups is the one that sends each abelian group in the corresponding zero ring.

Then the localization intuitively is just the ring obtained by inverting powers of f. If f is nilpotent, the localization is the zero ring.

In ring theory, a branch of mathematics, a zero ring is a ring (without unity) in which the product of any two elements is 0 (the additive identity element).

For this reason, it is often called the zero ring (not to be confused with a zero ring, although the trivial ring is a zero ring).

Another common name for zero rings is null ring; since the ring is not required to have unity, ideals can also be null rings, in which case they are referred to as null ideals.

The theory of finite rings is more complex than that of finite abelian groups, since any finite abelian group is the additive group of at least two nonisomorphic finite rings: the direct product of copies of , and the zero ring.