Statements (17)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:appliesTo |
modules over a ring
|
| gptkbp:consequence |
If f: M → M is a surjective endomorphism of a finitely generated module M over a local ring, then f is an isomorphism.
A finitely generated module over a local ring is zero if and only if its reduction modulo the maximal ideal is zero. |
| gptkbp:field |
gptkb:commutative_algebra
|
| gptkbp:firstPublished |
1940s
|
| gptkbp:generalizes |
gptkb:Krull's_intersection_theorem
|
| https://www.w3.org/2000/01/rdf-schema#label |
Nakayama's lemma
|
| gptkbp:namedAfter |
gptkb:Tadasi_Nakayama
|
| gptkbp:relatedTo |
gptkb:Jacobson_radical
|
| gptkbp:sentence |
If M is a finitely generated module over a local ring R with maximal ideal m, and mM = M, then M = 0.
If N is a submodule of a finitely generated module M such that M = N + mM, then M = N. |
| gptkbp:usedIn |
gptkb:algebraic_geometry
gptkb:commutative_algebra module theory |
| gptkbp:bfsParent |
gptkb:commutative_algebra
|
| gptkbp:bfsLayer |
5
|