Pseudoregular radical classes. (Q2777716)

Let \(\mathbb{Z}_0[x,y]\) denote the ring of integer polynomials in two variables without constant term. An element \(f\) of \(\mathbb{Z}_0[x,y]\) is called associating if there exists \(g\in\mathbb{Z}_0[x,y,z]\) such that \(f(f(x,y),z)=f(x,g(x,y,z))\). If \(f\) is associating then the class \({\mathcal R}_f\) of all rings \(R\) such that for all \(r\in R\) there exists \(s\in R\) such that \(f(r,s)=0\), is a radical class. In this paper the case that \(f(x,y)=p(x)+q(x)y\) for some \(p\in\mathbb{Z}_0[x]\), \(q(x)\in\mathbb{Z}[x]\) is considered. \({\mathcal R}_f\) will be denoted \({\mathcal R}_{p,q}\) in this case, and \((p,q)\) will be called an \(\mathcal R\)-pair.NEWLINENEWLINE Some results are given on \(\mathcal R\)-pairs and then attention is given to the radicals they define. It is shown that \({\mathcal R}_{p,q}\) contains all zero rings if and only if the coefficient of \(x\) in \(p(x)\) is a multiple of \(q(0)\). Two \(\mathcal R\)-pairs \((p,q)\) and \((p',q')\) are equivalent if \({\mathcal R}_{p,q}={\mathcal R}_{p',q'}\). An example of an \(\mathcal R\)-pair equivalent to a given \(\mathcal R\)-pair \((p,q)\) is given. Finally, some special cases are considered. A sufficient condition is shown for \(f(x,y)=x^n-(x^m-x^\ell)y\) to be associating. It is shown that if \(q(1)=0\) and \(p(1)=1\), then \({\mathcal R}_{p,q}\subseteq{\mathcal J}\), where \(\mathcal J\) denotes the Jacobson radical class. Finally the question as to when when does \({\mathcal R}_{p,q}\) contain all the nil rings is addressed, and a partial characterisation is given.