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:
“While the light burning within may have been divine, the outer case of the lamp was assuredly cheap enough. Whitman was, from first to last, a boorish, awkward poseur.”
—Rebecca Harding Davis (1831–1910)
“There are a great many of these accusers, and they have been accusing me now for a great many years, and what is more, they approached you at the most impressionable age, when some of you were children or adolescents; and literally won their case by default, because there was no one to defend me.”
—Socrates (469–399 B.C.)
“Ah, Christ, I love you rings to the wild sky
And I must think a little of the past:
When I was ten I told a stinking lie
That got a black boy whipped....”
—Allen Tate (1899–1979)