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\(P(r,m)\) near-rings. (English) Zbl 0990.16032
The \(P(r,m)\) near-rings of the title are right near-rings which satisfy the condition \(x^rN=Nx^m\) for all \(x\in N\), where \(r\) and \(m\) are positive integers. A number of examples of such near-rings are given. The properties and structure of these near-rings are investigated. The main conditions considered are \(S\)-near-rings (\(x\in Nx\) for all \(x\in N\)), \(S'\)-near-rings (\(x\in xN\) for all \(x\in N\)) and regularity. There are also strong connections with the absence of nilpotency and prime-like properties. There are many results connecting near-rings with these properties and showing that \(P(r,m)\) is quite a strong condition. To give a sample: an \(S'\)-near-ring \(N\) satisfying \(P(1,2)\) is subdirectly irreducible if and only if \(N\) is a near-field.
16Y30 Near-rings
16R99 Rings with polynomial identity
16N60 Prime and semiprime associative rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16U80 Generalizations of commutativity (associative rings and algebras)
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