The maximal orders which contain a given idealiser ring
Let \(C\) be a Dedekind domain, and let \(S\) be a maximal \(C\)-order in a simple Artinian ring \(Q\). Consider a proper essential right ideal \(K\) of \(S\) with \(SK=S\). The authors investigate the set \(\mathcal M\) of maximal \(C\)-orders in \(Q\) which contain the idealizer \(R\) of \(K\). They prove that the maximal \(C\)-orders in \(\mathcal M\) are in one-to-one correspondence with the right ideals \(I\) of \(S\) which satisfy \(K\subset I\) and \(RI=I\). If \(B\) denotes the bound of \(K\), that is, the largest two-sided ideal of \(S\) which is contained in \(K\), they make the further assumption that the right ideal \(K/B\) of \(S/B\) is generated by an idempotent element. (This always holds if \(R\) is hereditary.) Under this assumption they show that \(\mathcal M\) is finite, and that there is a bijection between \(\mathcal M\) and the set of ideals of \(S\) which contain \(B\). Moreover, they classify the invertible ideals of \(R\) and study their action on \(\mathcal M\) by conjugation. This generalizes classical results of Harada on hereditary orders over a Dedekind domain.
scientific article; zbMATH DE number 1539380