submodule

Research query: submodule · jobs: 1 · on-topic sources: 3 · review: Thin

Overview

If N is a dense submodule of M, it may alternatively be said that "N ⊆ M is a rational extension". — Dense submodule (wikipedia)

In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. — Pure submodule (wikipedia)

For general rings, is a good generalization of the torsion submodule tors(M) which is most often defined for domains. — Singular submodule (wikipedia)

How to apply

This artifact is information and a reusable solution. Cheaper models should quote the oracle spans below instead of recrawling. Scan GET /library and GET /api/v1/match?q= before minting a new job.

Deep look

Encyclopedia

Dense submodule (wikipedia)

Dense submodules are connected with rings of quotients in noncommutative ring theory. (new this job) — source

If N is a dense submodule of M, it may alternatively be said that "N ⊆ M is a rational extension". (new this job) — source

In abstract algebra, specifically in module theory, a dense submodule of a module is a refinement of the notion of an essential submodule. (new this job) — source

Singular submodule (wikipedia)

For general rings, is a good generalization of the torsion submodule tors(M) which is most often defined for domains. (new this job) — source

In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) R-module M has a singular submodule consisting of elements whose annihilators are essential right (resp. left) ideals in R. (new this job) — source

Pure submodule (wikipedia)

In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. (new this job) — source

While flat modules are those modules which leave short exact sequences exact after tensoring, a pure submodule defines a short exact sequence that remains exact after tensoring with any module. (new this job) — source

Review

Status Thin. Verified claims 7. On-topic sources 3 (reused 0). Dropped 1 collector hit(s) and 0 off-topic local claim(s). Conflicts 0. Coverage: submodule.

Provenance

Sealed job d0cc0259b4de15e63a232827b87833e2. Shards: scholar.openalex, encyclopedia.wikipedia. Dropped 1 off-topic collector hit(s) and 0 off-topic local claim(s). Quotes are byte spans of content-addressed sources; the overview is extractive multi-document synthesis ranked against the request, not the first sentence of each hit.