A class of rings for which the lattice of preradicals is not a set. (Q2920194)

In this paper all rings are associative and have identity. Let \(R\) be such a ring. \(\mathcal I(R_R)\) denotes the set of right ideals of \(R\), Mod-\(R\) stands for the category of all unital right \(R\)-modules and \(R\)-pr denotes the class of all preradicals over \(R\), that is, subfunctors of the identity functor on Mod-\(R\). A preradical \(\sigma\) is a radical if and only if \(\sigma(M/\sigma(M))=0\) for each \(M\in\text{Mod-}R\). A ring \(R\) is right hereditary if each \(J\in\mathcal I(R_R)\) is projective. Let \(M\in\text{Mod-}R\) and \(N\leq M\). \(N\) is called a radical submodule of \(M\) if there exists \(\sigma\in R\)-pr, \(\sigma\neq\widehat 1\), such that \(\sigma\) is a radical and \(N=\sigma(M)\). The authors call \(M\) a radical module if there exists \(L\in\text{Mod-}R\) such that \(M\) is a radical submodule of \(L\). They call a ring \(R\) a right (left) radical ring if the regular module \(R_R\), (\(_RR\)) is a radical module.NEWLINENEWLINE An interesting problem in the theory of preradicals is to describe the rings \(R\) for which the big lattice \(R\)-pr of preradicals over \(R\) is not a set. In this paper the authors show that for every right radical ring \(R\) the lattice \(R\)-pr is not a set. They also find a class of rings which are radical rings. For this purpose, for an integral domain \(Z\) the authors define a right \(Z\)-coinitial ring, namely, a ring \(R\) which is not a division ring, has as a subring of its centre a copy of \(Z\), and every nonzero right ideal contains an ideal of the form \(nR\) for some \(n\in Z\setminus\{0\}\). They describe commutative Noetherian domains and discrete valuation domains which are \(Z\)-coinitial. They prove that every \(Z\)-coinitial right hereditary countable ring \(R\) is a right radical ring. This enables the authors to present a list of examples which are right radical rings, so that \(R\)-pr is not a set.