Homological classification of rings
A general name for results which deduce properties of a ring (usually, an associative ring or a ring with a unit element) from the properties of certain modules over it — in particular, from the properties of the category of all left (or right) modules over this ring (cf. Morita equivalence; Module).
The following are the most important examples of such results.
2) A commutative local Noetherian ring is regular if and only if it has finite global homological dimension.
4) The projectivity of all flat left modules is equivalent to the minimum condition for principal right ideals (cf. Perfect ring).
5) A ring is left Noetherian if and only if the class of injective left modules over it may be described by formulas of first-order predicate calculus in the language of the theory of modules .
|||H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)|
|||J. Lambek, "Lectures on rings and modules" , Blaisdell (1966)|
|||L.A. Skornyakov, "Homological classification of rings" Mat. Vesnik , 4 : 4 (1967) pp. 415–434 (In Russian)|
|||P. Eklof, G. Sabbagh, "Model-completions and modules" Ann. Math. Logic , 2 : 3 (1971) pp. 251–295|
|||S. MacLane, "Homology" , Springer (1963)|
Homological classification of rings. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homological_classification_of_rings&oldid=39989