Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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A GENERALIZATION OF COHEN-MACAULAY MODULES BY TORSION THEORY

  • BIJAN-ZADEH, M.H. (Institute of Mathematics, University for Teacher Education) ;
  • PAYROVI, SH. (Institute of Mathematics, University for Teacher Education)
  • Received : 1997.07.11
  • Published : 1998.07.30
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Abstract

In this short note we study the torsion theories over a commutative ring R and discuss a relative dimension related to such theories for R-modules. Let ${\sigma}$ be a torsion functor and (T, F) be its corresponding partition of Spec(R). The concept of ${\sigma}$-Cohen Macaulay (abbr. ${\sigma}$-CM) module is defined and some of the main points concerning the usual Cohen-Macaulay modules are extended. In particular it is shown that if M is a non-zero ${\sigma}$-CM module over R and S is a multiplicatively closed subset of R such that, for all minimal element of T, $S{\cap}p={\emptyset}$, then $S^{-1}M$ is a $S^{-1}{\sigma}$-CM module over $S^{-1}$R, where $S^{-1}{\sigma}$ is the direct image of ${\sigma}$ under the natural ring homomorphism $R{\longrightarrow}S^{-1}R$.

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References

  1. J. London. Math. Soc. v.19 no.2 Torsion theories and local cohomology over commutative Noetherian rings Bijan-Zadeh, M.H.
  2. Local cohomology and localization, Pitman Research Notes in Mathematical Series No. 226 Bueso, J.L.;Torrecillas, B.;Verschoren, A.
  3. Proc. Amer. Math. Soc. v.38 Torsion theory and associated primes Cahen, P.J.
  4. Trans. Amer. Math. Soc. v.184 Commutative torsion theory Cahen, P.J.
  5. Math. Nachr. v.85 Verallgemeinerte Cohen-Macaulay Modulen Cuong, N.T.;Schenzel, P.;Trung, N.V.
  6. Quart. J. Math. Oxford v.23 no.2 An elementary proof of the non- vanishing of certain local cohomology modules Macdonald, I.G.;Sharp, R.Y.
  7. Trans. Amer. Math. Soc. v.97 Modules with descending chain condition Matlis, E.
  8. Commutative Ring Theory Matsumura, H.
  9. Math. Proc. Camb. Phil. Soc. v.107 On asymptotic stability for sets of prime ideals connected with the powers of an ideal Melkersson, L.
  10. Proc. Edinburgh Math. Soc. v.24 no.2 On the attached prime ideals of certain Artinian local cohomology modules Sharp, R.Y.