File:Module properties in commutative algebra.svg
Summary
| Description |
English: Properties of modules in commutative algebras and implications between them.
References: free => projective: projective modules are exactly direct summands of free ones: Lang III.4, p. 137. projective => flat: same reason, tensor products commute with direct sums. flat => torsion-free: torsion is the kernel of M tensor (A -> Q) if Q is the total quotient ring, M the module, A the ring. module torsion-free + ring Dedekind => module flat: Liu, Corollary 1.2.14 module flat and ring perfect => module projective: wikipedia page for perfect rings module projective + ring local => module free: Matsumura, Theorem 2.5module projective + ring PID => module free: Lang, Theorem III.7.1, p. 146 and App. 2.2, p. 880 |
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| Source | Own work |
| Author | KonradVoelkel |
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