# Homological classification of rings

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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].

#### 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)
How to Cite This Entry:
Homological classification of rings. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homological_classification_of_rings&oldid=39989
This article was adapted from an original article by A.V. MikhalevL.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article