Bounded And Fully Bounded Modules

Authors

  • ahmad haghany
  • majid mazrooei
  • mohammad reza vedadi

Keywords:

Bounded module, FBN ring, Krull dimension, $\mathcal{L}_2$-Noetherian module, $\mathcal{L}_2$-prime module

Abstract

Generalizing the concept of right bounded rings, a module $M_R$ is called bounded if {\rm ann}$_R(M/N)\leq_eR_R$ for all $N\leq_e M_R$. The module $M_R$ is called fully bounded if $(M/P)$ is bounded as a module over $R/{\rm ann}_R(M/P)$ for any $\mathcal{L}_2$-prime submodule $P\lhd M_R$. Bounded and right bounded are Morita invariant properties. Rings with all modules (fully) bounded are characterized and it is proved that a ring $R$ is right Artinian if and only if $R_R$ has Krull dimension, all $R$-modules are fully bounded and ideals of $R$ are finitely generated as right ideals. For certain fully bounded $\mathcal{L}_2$-Noetherian module $M_R$, it is shown that K.dim$(M_R)\leq$ Cl.K.dim$(R)$ when both dimensions exist. doi:10.1017/S0004972711002814

Published

2011-10-17

Issue

Vol. 84 No. 3 (2011)

Section

Articles
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