Multiplicative Identities and Commutativity Conditions

Abstract

We now have realized that semigroup theory can be useful for studying nearrings, so it is natural to examine nearrings whose multiplicative semigroup (or whose Clay semigroup) belongs to a given variety of semigroups. To begin (we will always treat only the initial part of various theories because at this point we will avoid delving into the more complex developments), we can study nearrings fulfilling some multiplicative identities.

As a preamble, we will discuss some of the questions (such as, for example, a generalization of a Putcha-Yaqub theorem) on semigroups to suggest the use of certain semigroup-theoretical invariants to classify and to study nearrings fulfilling multiplicative identities.

However, because we are not aware of any other in-depth studies on the general question of seeing when a semigroup is the multiplicative semigroup of a nearring, we will therefore go directly to the study of the nearrings N with certain weak distributivity conditions that imply conditions on the multiplicative semigroup [N;·].

More common studies are on a nearring N fulfilling given identities of small size, mainly in the cases in which N is simple or subdirectly irreducible.

Classical results, by Wedderburn, Jacobson and Herstein, giving sufficient conditions for having commutativity of a ring do not use such identities, but instead, they use less demanding conditions (such as x ^{n ( x )} = x, which is equivalent to ask that each x multiplicatively generates a finite group), usually absorbing the most important identities of small size (as the Boolean identity x ² = x). So, we will discuss the nearrings fulfilling the more interesting conditions (borrowed from classical papers) collected in definition 4.2.2, to also encounter simpler semigroup identities.

In particular, we will discuss potent and periodic nearrings, also using studies on the condition x—x ^{n (x)} ∈ Z (here called Herstein’s condition), and we will meet, as potent, all Boolean nearrings and, as periodic, all near-idempotent nearrings and all self-distributive nearrings. The last cases are simple enough to give simple sharp results. Also, conditions being similar to the classical (xy)^{n (x,y)} = xy are studied (under the name of PC-conditions).

To return to the historical source of these topics, we will have a taste of (multiplicative, but also additive) commutativity results on nearrings. Of course, a complete report cannot be given here because usually ring-theoretical techniques do not work on nearrings and even more complex discussions are necessary to get nontrivial results, so we (after the commutativity results obtained in studying previously quoted conditions) only give some commutativity results that were obtained by classical conditions on commutators and by discussions on derivation and α-derivations.

Una ricerca matematica non può mai essere considerata conclusa.

—Alessandro Terracini