The endomorphism ring of a bounded Abelian \(p\)-group (Q2738745)

Let \(G\) be a bounded Abelian \(p\)-group of exponent \(p^n\), let \({\mathcal E}(G)\) be its endomorphism ring, and denote the Jacobson radical of \({\mathcal E}(G)\) by \(\mathcal J\). It is well known that \(\mathcal J\) coincides with the nilpotent radical of \({\mathcal E}(G)\), and \({\mathcal J}^n=0\). The Loewy sequence of \(\mathcal J\) is the descending sequence \(({\mathcal J}^k)\), and \(\lambda\) denotes its length. Thus, \(\lambda\leq n\). \textit{C. E. Praeger} and \textit{P. Schultz} introduced the digraph, \(D(G)\), of \(G\) as a directed graph determined only by the integers \(k\) for which \(G\) has a nonzero Ulm invariant \(f_k(G)\) and gave a formula for \(\lambda\) in terms of the length of the longest path in \(D(G)\) [Abelian groups and noncommutative rings, Contemp. Math. 130, 349-360 (1992; Zbl 0773.20024)]. In the current article, the authors classify the ideals of \({\mathcal E}(G)\) by numerical invariants and apply their results to determine the upper annihilator sequence and the Loewy sequence of \(\mathcal J\).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].