Mathieu-Zhao spaces of univariate polynomial rings with non-zero strong radical (Q273011)

Let \(k\) be a field of characteristic \(0\), \(R\) a commutative \(k\)-algebra and \(V\subset R\) a \(k\)-linear subspace of \(R\) (not necessarily an ideal). The authors define some kinds of radicals of \(V\): NEWLINE\[NEWLINE \begin{aligned} r(V) :&=\{a\in R:{\exists }_{\substack{ N }} a^{m}\in V\text{ for }m\geq N\}, \\ sr(V) :&=\{a\in R:{\forall }_{b\in R}~{\exists}_{N_{b}}ba^{m}\in V\text{ for }m\geq N_{b}\}. \end{aligned} NEWLINE\]NEWLINE \(V\) is called a Mathieu-Zhao subspace of \(R\) if \(r(V)=sr(V)\) (always \(r(V)\supset sr(V))\). The main result of the paper is a characterization of the Mathieu-Zhao subspaces \(V\) of the univariate polynomial ring \(k[t]\) for which \(sr(V)\neq \varnothing\).