Commun. Korean Math. Soc. 2018; 33(3): 711-719
Online first article February 26, 2018 Printed July 31, 2018
https://doi.org/10.4134/CKMS.c170276
Copyright © The Korean Mathematical Society.
Bahram Amirsardari, Saeid Bagheri
Malayer University, Malayer University
In this paper we define and study a new kind of dimension called, semisimple dimension, that measures how far a module is from being semisimple. Like other kinds of dimensions, this is an ordinal valued invariant. We give some interesting and useful properties of rings or modules which have semisimple dimension. It is shown that a noetherian module with semisimple dimension is an artinian module. A domain with semisimple dimension is a division ring. Also, for a semiprime right non-singular ring $R$, if its maximal right quotient ring has semisimple dimension as a right $R$-module, then $R$ is a semisimple artinian ring. We also characterize rings whose modules have semisimple dimension. In fact, it is shown that all right $R$-modules have semisimple dimension if and only if the free right $R$-module $\oplus_{i=1}^{\infty}R$ has semisimple dimension, if and only if $R$ is a semisimple artinian ring.
Keywords: uniform dimension, semisimple module, artinian module
MSC numbers: 16D90, 16P70, 03E10
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