{"paper":{"title":"Submodule Categories of Wild Representation Type","license":"","headline":"","cross_cats":["math.CT"],"primary_cat":"math.RT","authors_text":"Claus Michael Ringel, Markus Schmidmeier","submitted_at":"2004-09-21T22:51:01Z","abstract_excerpt":"Let $\\Lambda$ be a commutative local uniserial ring of length at least seven with radical factor ring $k$. We consider the category $S(\\Lambda)$ of all possible embeddings of submodules of finitely generated $\\Lambda$-modules and show that $S(\\Lambda)$ is controlled $k$-wild with a single control object $I\\in S(\\Lambda)$. In particular, it follows that each finite dimensional $k$-algebra can be realized as a quotient $\\End(X)/\\End(X)_I$ of the endomorphism ring of some object $X\\in S(\\Lambda)$ modulo the ideal $\\End(X)_I$ of all maps which factor through a finite direct sum of copies of $I$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0409417","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"}