Math 2.0

Associative rings and algebras Local rings and generalizations Theorem

Bass's theorem P

Builds on RingJacobson radicalPerfect ringProjective coverProjective module

Statement

Let RR be a ring with identity and J=J(R)J = J(R) its Jacobson radical. The following are equivalent.

  1. R/JR/J is a semisimple ring and JJ is left T-nilpotent, that is, RR is left perfect.
  2. Every left RR-module has a projective cover.
  3. RR satisfies the descending chain condition on principal right ideals.
  4. Every flat left RR-module is projective.

Remarks

The link between 1 and 2 goes through Nakayama-type lemmas. Left T-nilpotence of JJ is exactly what makes JM≠MJM \ne M for every nonzero left module MM, finitely generated or not. Then a projective cover of MM can be built by lifting a projective cover of the semisimple module M/JMM/JM. Condition 3 is notable for its side switch: a condition on left modules is equivalent to a chain condition on right ideals. Example: left Artinian rings satisfy all four conditions, so over them every left module has a projective cover and flat left modules are projective. Over ℤ\Z, which is not perfect, the flat module ℚ\Q is not projective, and the module ℤ/2ℤ\Z/2\Z has no projective cover.

Going deeper

For a student

The theorem is the main characterization of perfect rings and appears in the study of projective covers and flat modules. Its finitely generated analogue characterizes semiperfect rings: every finitely generated left module has a projective cover exactly when RR is semiperfect. Because semiprimary rings have nilpotent radical, they satisfy condition 1 on both sides.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

Show the path as a graph
Commutative ring, concept, no evidence levelCommutative ringconceptExact sequence, concept, no evidence levelExact sequenceconceptFree module, concept, no evidence levelFree moduleconceptGroup, concept, no evidence levelGroupconceptJacobson radical, concept, no evidence levelJacobson radicalconceptLinear map, concept, no evidence levelLinear mapconceptModule over a ring, concept, no evidence levelModule over a ringconceptProjective cover, concept, no evidence levelProjective coverconceptSemisimple ring, concept, no evidence levelSemisimple ringconceptProjective module, definition, no evidence levelProjective moduledefinitionRing, definition, no evidence levelRingdefinitionSemilocal ring, definition, no evidence levelSemilocal ringdefinitionPerfect ring, definition, no evidence levelPerfect ringdefinitionBass's theorem P, statement, no evidence levelBass's theorem Pstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Commutative ring concept
  2. Exact sequence concept
  3. Free module concept
  4. Group concept
  5. Jacobson radical concept
  6. Linear map concept
  7. Module over a ring concept
  8. Projective cover concept
  9. Semisimple ring concept
  10. Projective module definition
  11. Ring definition
  12. Semilocal ring definition
  13. Perfect ring definition
  14. Bass's theorem P statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Perfect_ring
About this page
TagD2M8BSW720
URIm2://D2M8BSW720
Kindstatement, theorem
WritingW0 stub
MSC16L30, 16D40
StatusThis copy of the site has no build data. Run m2 status D2M8BSW720.
AboutPerfect ring
UsesRing, Jacobson radical, Perfect ring, Projective cover, Projective module

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