Abstract
A direct product decomposition is given for the multiplicative semigroup of a finite near integral domain in terms of the subsemigroup of left identities and a group of automorphisms on the additive group of the domain. Conditions are given which insure that every element will have a uniquen-th root. If there existsx≠0 such that (−x)y=−(xy), for eachy, then the additive group of the near integral domain is abelian. Other conditions sufficient for the commutativity of the additive group are given. An example illustrates that non-isomorphic finite near integral domains can have a left ideal decomposition into Sylow subgroups which are isomorphic as near-rings. Another example shows that an infinite near integral domain need not have a nilpotent additive group, even in the d. g. case. It is conjectured that for each natural numbern there is a near integral domain whose additive group is of nilpotent classn.
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