- Volume 33 Issue 3
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SEMISIMPLE DIMENSION OF MODULES
- Amirsardari, Bahram (Department of Mathematics Malayer University) ;
- Bagheri, Saeid (Department of Mathematics Malayer University)
- Received : 2017.07.05
- Accepted : 2017.09.14
- Published : 2018.07.31
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
uniform dimension;semisimple module;artinian module
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