Every zero ring is a nilpotent ring. For if is a zero ring, then .
Zero rings exist in . 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 (http://planetmath.org/FundamentalTheoremOfFinitelyGeneratedAbelianGroups). Thus, if are distinct primes, are positive integers, and , then the number of zero rings of order (http://planetmath.org/Order) is , where denotes the partition function (http://planetmath.org/PartitionFunction2).
|Date of creation||2013-03-22 13:30:19|
|Last modified on||2013-03-22 13:30:19|
|Last modified by||Wkbj79 (1863)|