# SEMISIMPLE DIMENSION OF MODULES

• Amirsardari, Bahram (Department of Mathematics Malayer University) ;
• Bagheri, Saeid (Department of Mathematics Malayer University)
• Accepted : 2017.09.14
• Published : 2018.07.31
• 173 13

#### Abstract

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 ${\oplus}^{\infty}_{i=1}$ R has semisimple dimension, if and only if R is a semisimple artinian ring.

#### Keywords

uniform dimension;semisimple module;artinian module

#### References

1. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, second edition, Graduate Texts in Mathematics, 13, Springer-Verlag, New York, 1992.
2. P. Aydogdu, A. C. Ozcan, and P. F. Smith, Chain conditions on non-summands, J. Algebra Appl. 10 (2011), no. 3, 475-489. https://doi.org/10.1142/S0219498811004707
3. A. Ghorbani, S. K. Jain, and Z. Nazemian, Indecomposable decomposition and couniserial dimension, Bull. Math. Sci. 5 (2015), no. 1, 121-136. https://doi.org/10.1007/s13373-014-0062-6
4. K. R. Goodearl, Ring Theory, Marcel Dekker, Inc., New York, 1976.
5. K. R. Goodearl and R. B. Warﬁeld, Jr., An Introduction to Noncommutative Noether- ian Rings, second edition, London Mathematical Society Student Texts, 61, Cambridge University Press, Cambridge, 2004.
6. T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 1991.
7. Z. Nazemian, A. Ghorbani, and M. Behboodi, Uniserial dimension of modules, J. Algebra 399 (2014), 894-903. https://doi.org/10.1016/j.jalgebra.2013.09.054
8. R. R. Stoll, Set Theory and Logic, corrected reprint of the 1963 edition, Dover Publications, Inc., New York, 1979.
9. R. Wisbauer, Foundations of Module and Ring Theory, revised and translated from the 1988 German edition, Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.