A study on annihilator conditions of polynomials (Q2712546)

Let \(G\) be a group and \(M_0(G)=\{f\colon G\to G\mid f(0)=0\}\) the near-ring of all zero preserving mappings on \(G\). First, it is shown that \(M_0(G)\) is a Baer near-ring. From this result, it is proved that every zero-symmetric near-ring can be embedded into a Baer near-ring. It is also proved that the zero-symmetric part of the polynomial near-ring \(R[x]\) is a Baer (resp. p.p.) near-ring if and only if \(R\) is a Baer (resp. p.p.) ring. Moreover the author obtains that, if \(R\) is an associative ring with identity and \(M\) is a unital left \(R\)-module, then \(R\oplus M\) is a p.p. near-ring if and only if \(R\) is a p.p. ring (the notion of ``p.p. ring'' was introduced by \textit{A. Hattori} [in Nagoya Math. J. 17, 147-158 (1960; Zbl 0117.02202)]).