On normal radicals. (Q2896991)

Let \(R\) and \(S\) be semirings, \(V={_RV_S}\) an \(R\)-\(S\)-bisemimodule, \(W={_SW_R}\) an \(S\)-\(R\)-bisemimodule, and assume that there are mappings \(V\times W\to R\) and \(W\times V\to S\) such that the identities \((v_1+v_2)w=v_1w+v_2w\), \(r(vw)=(rv)w\), \((vs)w=v(sw)\), \(v(w_1+w_2)=vw_1+vw_2\), \((vw)r=v(wr)\), \((v_1w)v_2=v_1(wv_2)\) and their duals are satisfied for all \(v,v_i\in V\), \(w,w_i\in W\), \(r\in R\) and \(s\in S\). Then \((R,V,W,S)\) is called a Morita context, and the set \(\left\{\left(\begin{smallmatrix} r&v\\ w&s\end{smallmatrix}\right)\mid r\in R,\;s\in S,\;v\in V,\;w\in W\right\}\) of matrices is a semiring. Let \(\mathcal R\) be a radical class and \(\mathcal R(R)\) and \(\mathcal R(S)\) the radicals of \(R\) and \(S\), respectively. Then \(\mathcal R\) is called normal if \(V\mathcal R(S)W\subseteq\mathcal R(R)\) for every Morita context \((R,V,W,S)\). Some characterizations of normal radical classes and properties of such classes are proved.