Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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INDEPENDENTLY GENERATED MODULES

  • Kosan, Muhammet Tamer (DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE GEBZE INSTITUTE OF TECHNOLOGY) ;
  • Ozdin, Tufan (DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE AND LITERATURE ERZINCAN UNIVERSITY)
  • Published : 2009.09.30
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Abstract

A module M over a ring R is said to satisfy (P) if every generating set of M contains an independent generating set. The following results are proved; (1) Let $\tau$ = ($\mathbb{T}_\tau,\;\mathbb{F}_\tau$) be a hereditary torsion theory such that $\mathbb{T}_\tau$ $\neq$ Mod-R. Then every $\tau$-torsionfree R-module satisfies (P) if and only if S = R/$\tau$(R) is a division ring. (2) Let $\mathcal{K}$ be a hereditary pre-torsion class of modules. Then every module in $\mathcal{K}$ satisfies (P) if and only if either $\mathcal{K}$ = {0} or S = R/$Soc_\mathcal{K}$(R) is a division ring, where $Soc_\mathcal{K}$(R) = $\cap${I 4\leq$ $R_R$ : R/I$\in\mathcal{K}$}.

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References

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