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.

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 [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 [2].

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 [4].

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

References

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