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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.
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)
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