Quasi-complete Semilocal Rings and Modules



Let R be a (commutative Noetherian) semilocal ring with Jacobon radical J. Chevalley has shown that if R is complete, then R satisfies the following condition: given any descending chain of ideals \(\left \{A_{n}\right \}_{n=1}^{\infty }\) with \(\bigcap \nolimits _{n=1}^{\infty }A_{n} = 0\), for each positive integer k there exists an s k with \(A_{s_{k}} \subseteq J^{k}\). A finitely generated R-module M is said to be (weakly) quasi-complete if for any descending chain \(\left \{A_{n}\right \}_{n=1}^{\infty }\) of R-submodules of M (with \(\bigcap \nolimits _{n=1}^{\infty }A_{n} = 0\)) and k ≥ 1, there exists an s k with \(A_{s_{k}} \subseteq (\bigcap \nolimits _{n=1}^{\infty }A_{n}) + J^{k}M\). An easy modification of Chevalley’s proof shows that a finitely generated R-module over a complete semilocal ring is quasi-complete. However, the converse is false as any DVR is quasi-complete. In this paper we survey known results about (weakly) quasi-complete rings and modules and prove some new results.


Quasi-complete rings Quasi-complete modules Noether lattices 

Subject Classifications

13E05 13H10 13A15 06F10 


  1. 1.
    D.D. Anderson, Multiplication ideals, multiplication rings, and the ring \(R\left (X\right )\). Canad. J. Math. 27, 760–768 (1976)CrossRefGoogle Scholar
  2. 2.
    D.D. Anderson, The existence of dual modules. Proc. Am. Math. Soc. 55, 258–260 (1976)CrossRefzbMATHGoogle Scholar
  3. 3.
    D.D. Anderson, M. Axtell, S.J. Forman, J. Stickles, When are associates unit multiples? Rocky Mount. J. Math. 34, 811–823 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    A. Barnard, Multiplication modules. J. Algebra 71, 174–178 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    N. Bourbaki, Commutative Algebra (Addison-Wesley Publishing Company, Reading, 1972)zbMATHGoogle Scholar
  6. 6.
    C. Chevalley, On the theory of local rings. Ann. Math. 44, 690–708 (1943)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    R.P. Dilworth, Abstract commutative ideal theory. Pacific J. Math. 12, 481–498 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    E.W. Johnson, A note on quasi-complete local rings. Coll. Math. 21, 197–198 (1970)zbMATHGoogle Scholar
  9. 9.
    E.W. Johnson, Modules: duals and principally generated fake duals. Algebra Universalis 24, 111–119 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    E.W. Johnson, J.A. Johnson, The Hausdorff completion of the space of closed subsets of a module. Canad. Math. Bull. 38, 325–329 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    J.A. Johnson, a-adic completions of Noetherian lattice modules. Fund. Math. 66, 347–373 (1970)Google Scholar
  12. 12.
    J.A. Johnson, Semi-local lattices. Fund. Math. 90, 11–15 (1975)zbMATHGoogle Scholar
  13. 13.
    J.A. Johnson, Quasi-complete ideal lattices. Coll. Math. 33, 59–62 (1975)zbMATHGoogle Scholar
  14. 14.
    J.A. Johnson, Completeness in semilocal ideal lattices. Czechoslovak Math. J. 27, 378–387 (1977)MathSciNetGoogle Scholar
  15. 15.
    J.A. Johnson, Quasi-completeness in local rings. Math. Japon. 22, 183–184 (1977)zbMATHMathSciNetGoogle Scholar
  16. 16.
    C.-P. Lu, Quasi-complete modules. Indiana Univ. Math. J. 29, 277–286 (1980)CrossRefzbMATHGoogle Scholar
  17. 17.
    M. Nagata, Local Rings, Interscience Tract in Pure and Applied Mathematics, vol. 13 (Interscience, New York, 1962)Google Scholar
  18. 18.
    D.G. Northcott, Ideal Theory (Cambridge University Press, Cambridge, 1953)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA

Personalised recommendations