{"paper":{"title":"On the arithmetic of the endomorphism ring End($\\mathbb{Z}_{p}\\times\\mathbb{Z}_{p^{m}}$)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hualu Liu, Xiusheng Liu","submitted_at":"2016-05-03T09:27:51Z","abstract_excerpt":"For a prime $p$, let $E_{p,p^m}=\\{\\begin{pmatrix}a&b\\\\p^{m-1}c&d\\end{pmatrix}|a,b,c\\in\\mathbb{Z}_{p},~\\mathrm{and}~d\\in \\mathbb{Z}_{p^{m}}\\}$. We first establish a ring isomorphism from $\\mathrm{End}(\\mathbb{Z}_p\\times\\mathbb{Z}_p^m)$ onto $E_{p,p^m}$. We then provide the way to compute $-d$ and $d^{-1}$ using arithmetic in $\\mathbb{Z}_{p}$ and $\\mathbb{Z}_{p^{m}}$, and characterize invertible elements in $E_{p,p^m}$. Moreover, we introduce the minimal polynomial for each element in $E_{p,p^m}$ and given its applications."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.00805","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}