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:
“I often used to think myself in the case of the fox-hunter, who, when he had toiled and sweated all day in the chase as if some unheard-of blessing was to crown his success, finds at last all he has got by his labor is a stinking nauseous animal. But my condition was yet worse than his; for he leaves the loathsome wretch to be torn by his hounds, whilst I was obliged to fondle mine, and meanly pretend him to be the object of my love.”
—Sarah Fielding (1710–1768)
“Before I get through with you, you will have a clear case for divorce and so will my wife. Now, the first thing to do is arrange for a settlement. You take the children, your husband takes the house, Junior burns down the house, you take the insurance and I take you!”
—S.J. Perelman, U.S. screenwriter, Arthur Sheekman, Will Johnstone, and Norman Z. McLeod. Groucho Marx, Monkey Business, terms for a divorce settlement proposed while trying to woo Lucille Briggs (Thelma Todd)
“The next time the novelist rings the bell I will not stir though the meeting-house burn down.”
—Henry David Thoreau (1817–1862)