\(P(r,m)\) near-rings (Q5952801)

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.