Zero ring
Encyclopedia
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). (Note: Some authors
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 group
s of prime order.
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a zero ring is a ring (without unity) in which the product of any two elements is 0 (the additive identity element). (Note: Some authors
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
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
can also be null rings, in which case they are referred to as null ideals.
Any abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
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
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of the additive group of a zero ring is an ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
. It follows that the only zero rings that are simple
Simple ring
In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...
are those whose additive groups are cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
s of prime order.