Balakrishnan, R.; Suryanarayanan, S. \(P(r,m)\) near-rings. (English) Zbl 0990.16032 Bull. Malays. Math. Sci. Soc. (2) 23, No. 2, 117-130 (2000). 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. Reviewer: J.D.P.Meldrum (Edinburgh) Cited in 2 Documents MSC: 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) Keywords:\(P(r,m)\) near-rings; right near-rings; regularity; nilpotency; subdirectly irreducible near-rings; near-fields PDFBibTeX XMLCite \textit{R. Balakrishnan} and \textit{S. Suryanarayanan}, Bull. Malays. Math. Sci. Soc. (2) 23, No. 2, 117--130 (2000; Zbl 0990.16032) Full Text: EuDML