On annihilators of polynomial near-rings.
For a ring \(R\), let \((R[x],+,\cdot)\) denote the ring of polynomials in the indeterminate \(x\) over \(R\). If the usual multiplication \(\cdot\) of polynomials is replaced by composition \(\circ\) of polynomials, a near-ring \((R[x],+,\circ)\) is obtained. As the authors prefer to write the functions on the right of the elements they act on (so that \((x)g\circ (x)f=((x)g)f\)), the near-ring thus obtained is a structure that does not satisfy the right distributive law and is hence called a `left' near-ring (as it still satisfies the left distributive law). The subnear-ring that consists of all those polynomials with zero constant term is denoted by \(R_0[x]\). A ring \(R\) is called `near-Armendariz' if, for each pair of polynomials \((x)f=\sum_{i=1}^m f_ix^i\) and \((x)g=\sum_{j=1}^n g_jx^j\) in \(R_0[x]\), \((x)g\circ (x)f=0\) implies that \(f_ig_j=0\) for all \(1\leq i\leq m\) and \(1\leq j\leq n\). Similarly, \(R\) is called `quasi-near-Armendariz' if for each such pair \((x)f\) and \((x)g\), \((x)g\circ R_0[x]\circ (x)f=\{0\}\) implies that \(f_iRg_j=\{0\}\) for all \(1\leq i\leq m\) and \(1\leq j\leq n\). The authors show an example of a near-Armendariz ring (without left identity) that is not a reduced ring. (It was shown by \textit{G. F. Birkenmeier} and \textit{F.-k. Huang} [in Commun. Algebra 29, No. 5, 2097-2112 (2001; Zbl 0990.16033); Erratum 30, No. 7, 3565 (2002)] that every reduced ring is near-Armendariz.) Furthermore, they prove that a near-Armendariz ring with a left identity must be a reduced ring. Finally, quasi-near-Armendariz rings are characterized in terms of right and left annihilators of ideals of \(R\). It is found, for example, that all semiprime rings are quasi-near-Armendariz. Also, for any quasi-near-Armendariz ring \(R\), the ring \(eRe\) is also quasi-near-Armendariz, where \(e\) is any nonzero idempotent of \(R\).
scientific article; zbMATH DE number 5980778