View source for 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|Morita equivalence]]; [[Module|Module]]).

The following are the most important examples of such results.

1) The classical semi-simplicity of a ring is equivalent both to the injectivity of all left modules over it and to their projectivity, and also to the injectivity of all left ideals of the ring [[#References|[1]]].

2) A commutative local Noetherian ring is regular if and only if it has finite global homological dimension.

3) A ring is regular (in the sense of von Neumann) if and only if all modules over it are flat, i.e. if the ring has weak homological dimension zero [[#References|[2]]].

4) The projectivity of all flat left modules is equivalent to the minimum condition for principal right ideals (cf. [[Perfect ring|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 [[#References|[4]]].

See also [[Artinian ring|Artinian ring]]; [[Quasi-Frobenius ring|Quasi-Frobenius ring]]; [[Coherent ring|Coherent ring]]; [[Semi-perfect ring|Semi-perfect ring]]; [[Self-injective ring|Self-injective ring]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,   S. Eilenberg,   "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Lambek,   "Lectures on rings and modules" , Blaisdell  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. Skornyakov,   "Homological classification of rings"  ''Mat. Vesnik'' , '''4''' :  4  (1967)  pp. 415–434  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Eklof,   G. Sabbagh,   "Model-completions and modules"  ''Ann. Math. Logic'' , '''2''' :  3  (1971)  pp. 251–295</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. MacLane,   "Homology" , Springer  (1963)</TD></TR></table>