Case of Commutative Rings
For any multiplicatively closed subset S of R, the localization ring is flat as an R-module.
When R is Noetherian and M is a finitely-generated R-module, being flat is the same as being locally free in the following sense: M is a flat R-module if and only if for every prime ideal (or even just for every maximal ideal) P of R, the localization is free as a module over the localization .
If S is an R-algebra, i.e., we have a homomorphism, then S has the structure of an R-module, and hence it makes sense to ask if S is flat over R. If this is the case, then S is faithfully flat over R if and only if every prime ideal of R is the inverse image under f of a prime ideal in S. In other words, if and only if the induced map is surjective.
Flat modules over commutative rings are always torsion-free. Projective modules (and thus free modules) are always flat. For certain common classes of rings, these statements can be reversed (for example, every torsion-free module over a Dedekind ring is automatically flat and flat modules over perfect rings are always projective), as is subsumed in the following diagram of module properties:
Read more about this topic: Flat Module
Famous quotes containing the words case of, case and/or rings:
“In the case of our main stock of well-worn predicates, I submit that the judgment of projectibility has derived from the habitual projection, rather than the habitual projection from the judgment of projectibility. The reason why only the right predicates happen so luckily to have become well entrenched is just that the well entrenched predicates have thereby become the right ones.”
—Nelson Goodman (b. 1906)
“When a cat cries over a rat, it’s a case of false compassion.”
—Chinese proverb.
“The next time the novelist rings the bell I will not stir though the meeting-house burn down.”
—Henry David Thoreau (1817–1862)