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)^[1]. (Note: Some authors^[2] define a "zero ring" to be a ring with a single element, see trivial ring. These authors require their rings to have unity, hence all zero rings are trivial.) 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.

Any abelian group can be turned into a zero ring by defining the product of any two elements to be 0. This proves that any abelian group is the additive group of some ring.

Any subgroup of the additive group of a zero ring is an ideal. It follows that the only zero rings that are simple are those whose additive groups are cyclic groups of prime order^[3].

References

1. ^ Bourbaki, Nicolas (1970) (in fr). Algèbre (Chapitres 1 à 3). Hermann. p. I-97.  with the denomination pseudo-anneau de carré nul
2. ^ For example Warner, Seth (1990). Modern Algebra. 1. Courier Dover. p. 188. ISBN 0-486-66341-8. 
3. ^ Zariski, Oscar; Samuel, Pierre (1958). Commutative Algebra. 1. Van Nostrand. p. 133. 

• Hall, Frederick Michael (1969). An Introduction to Abstract Algebra. 2. CUP Archive. p. 64. 

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