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INJECTIVE COVERS OVER COMMUTATIVE NOETHERIAN RINGS OF GLOBAL DIMENSION AT MOST TWO II

KIM, HAE-SIK;SONG, YEONG-MOO

  • Published : 2005.07.01

Abstract

In studying injective covers, the modules C such that Hom(E, C) = 0 and $Ext^1$(E, C) = 0 for all injective module E play an important role because of Wakamatsu's lemma. If C is a module over the ring k[[x, y]] with k a field, the class of these modules C contains the class $\={D}$ of all direct summands of products of modules of finite length ([3, Theorem 2.9]). In this paper we show that every module over any commutative ring has a $\={D}$-preenvelope.

Keywords

injective cover

References

  1. D. Eisenbud, Commutative Algebra with a view toward algebraic geometry, Grad. Texts in Math. 1994
  2. E. Enochs, Injective and flat covers, envelopes, and resolvents, Israel J. Math. 39 (1981), 189-209 https://doi.org/10.1007/BF02760849
  3. E. Enochs, H. Kim, and Y. Song, Injective Covers over Commutative Noetherian Rings with Global Dimension at most two, Bull. Korean Math. Soc. 40 (2003), 167-176 https://doi.org/10.4134/BKMS.2003.40.1.167
  4. T. Wakamatsu, Stable equivalence of self-injective algebras and a generalization of tilting modules, J. Algebra 134 (1990), 298-325 https://doi.org/10.1016/0021-8693(90)90055-S
  5. J. Xu, Flat Covers of Modules, Lecture Notes in Math. vol. 1634, Springer-Verlag, Berlin, 1996