Maximal right nearring of quotients and semigroup generalized polynomial identity
Quotient near-rings have been studied before by several authors, mainly using a type of Ore condition. The approach in this paper is different. The near-rings considered are right zero-symmetric near-rings. Such a near-ring \(N\) is 3-semiprime if \(xNx\neq 0\) for any non-zero element \(x\in N\). It is equiprime if for any \(a,b,c\in N\), with \(a\neq 0\) the condition \(axc= axb\) for all \(x\in N\) implies \(b= c\). The present approach derives from \textit{Y. Utumi} [Osaka Math. J. 8, 1--18 (1956; Zbl 0070.26601)] as developed in \textit{K. I. Beidar}, \textit{W. S. Martindale III} and \textit{A. V. Mikhalev} [Rings with generalized identities, Marcel Dekker, New York (1996; Zbl 0847.16001)]. The approach is to consider the set \(D(N)\) of all dense right \(N\) invariant subsets of \(N\), that is subsets \(M\) of \(N\) invariant under multiplication on the right by \(N\) such that if \(x, y\in N\) with \(x\neq 0\) there exists \(z\in N\) such that \(xz\neq 0\) and \(yz\in M\). The set of pairs \((f, J)\) is defined where \(J\in D(N)\) and \(f\) is an \(N\) homomorphism from \(J\) to \(N\), the action of \(N\) on \(J\) and \(N\) being right multiplication. By taking suitable equivalence classes of such pairs and defining addition and multiplication on them, the maximal near-ring of quotients \({\mathcal Q}_{mr}(N)\) is obtained. It contains \(N\), has a unity and is unique up to isomorphism. If \(N\) is 3-semiprime, 3-prime or equiprime, then so is \({\mathcal Q}\). The process of constructing \({\mathcal Q}(N)\) is idempotent. The near-ring \({\mathcal Q}(N)\) is von Neumann regular if and only if \(N\) is right non-singular, that is for any element \(x\in N\) there exists a non-zero right \(N\) subset \(Z_x\) such that \(xy\neq xz\) for all \(y, z\in Z_x\) with \(y\neq z\). If \(N\) is zero-symmetric, 3-prime with a minimal left \(N\)-subgroup \(L\) such that \(L= Ne\) for an idempotent \(e\), then \(G= eNe\setminus\{0\}\) is a fixed point free group of automorphisms of \((L, +)\), and \({\mathcal Q}(N)\) is isomorphic to the centralizer near-ring \(M_G(L)\), where \({\mathcal Q}\) acts on \(L\) by left multiplication. There are several more results along these lines and an example to show that \(R\) can be a prime ring with its maximal right ring of quotients distinct from its maximal right near-ring of quotients. The final section considers the Martindale centroid of \(N\) (the mulplicative centre of \({\mathcal Q}(N)\)) and multilinear semigroup generalized identities, the precise definition of which is rather lengthy. The authors are able to show, among other things, that an equiprime near-ring satisfies a nontrivial multilinear semigroup generalized identity if and only if it is a centralizer near-ring determined by a commutative group of fixed point free automorphisms on an additive group. As is apparent, there are a substantial number of interesting results in this paper.
scientific article; zbMATH DE number 1997667
scientific article