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zero ring∗
Wkbj79†
2013-03-21 15:45:41
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 Z is a zero ring, 0Z
is its additive identity, and x, y ∈ Z, then xy = 0Z = yx.
Every zero ring is a nilpotent ring. For if Z is a zero ring, then Z 2 = {0Z }.
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 Z1 = {0}. 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 R
is a ring, then
r −r r
∈
R
r −r considered as a subring of M2 x 2 (R) (with standard matrix addition and multiplication) is a zero ring. Moreover, the cardinality of this subset of M2 x 2 (R)
is the same as that of R.
Moreover, zero rings can be constructed from any abelian group. If G is a
group with identity eG , it can be made into a zero ring by declaring its addition
to be its group operation and defining its multiplication by a · b = eG for any
a, b ∈ G.
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 p1 , . . . , pm are distinct primes, a1 , . . . , am are
m
Y
positive integers, and n =
pj aj , then the number of zero rings of order n is
j=1
m
Y
p(aj ), where p denotes the partition function.
j=1
∗ hZeroRingi
created: h2013-03-21i by: hWkbj79i version: h34086i Privacy setting: h1i
hDefinitioni h16U99i h13M05i h13A99i
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