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Difference between revisions of "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|Morita equivalence]]; [[Module|Module]]).
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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.
 
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]]].
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1) The [[Classical semi-simple ring|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.
 
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]]].
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3) A ring is [[Regular ring (in the sense of von Neumann)|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]]).
 
4) The projectivity of all flat left modules is equivalent to the minimum condition for principal right ideals (cf. [[Perfect ring|Perfect ring]]).
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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]]].
 
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]].
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See also [[Artinian ring]]; [[Quasi-Frobenius ring]]; [[Coherent ring]]; [[Semi-perfect ring]]; [[Self-injective ring]].
  
 
====References====
 
====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>
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<table>
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<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>
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<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>
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<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>
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<TR><TD valign="top">[5]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR>
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</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 07:32, 13 December 2016

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)
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