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The Krull Dimension of Certain Semiprime Modules Versus Their \(\alpha \)-Shortness

  • S. M. Javdannezhad
  • N. Shirali
Article
  • 17 Downloads

Abstract

We study the R-modules M which are finitely generated, quasi-projective and self-generator (briefly called \({{\mathrm{FQS}}}\) modules). We extend some basic results from semiprime rings to semiprime \({{\mathrm{FQS}}}\) modules. In particular, we show that any semiprime \({{\mathrm{FQS}}}\) module with Krull dimension is a Goldie module. We also show that every \({{\mathrm{FQS}}}\) module with Krull dimension has only finitely many minimal prime submodules. Consequently, if M is an \({{\mathrm{FQS}}}\) module with Krull dimension, then \(\text{ k-dim }\,M\) is equal to \(\text{ k-dim }\,\frac{M}{P}\) for some prime submodule P of M. Moreover, we observe that an \({{\mathrm{FQS}}}\) module has the classical Krull dimension if and only if it satisfies ACC on prime submodules. Finally, we prove that a semiprime \({{\mathrm{FQS}}}\) module M is \(\alpha \)-short if and only if \(\text{ n-dim }\, M =\alpha ,\) where \(\alpha \ge 0.\)

Keywords

\({{\mathrm{FQS}}}\) module Krull dimension classical Krull dimension 

Mathematics Subject Classification

Primary 16P60 16P20 Secondary 16P40 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShahid Chamran University of AhvazAhvazIran

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