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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.
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참고문헌 (9)
참고문헌 신청
F. W. Anderson / 1992 / Rings and Categories of Modules, second edition, Graduate Texts in Mathematics / Springer-Verlag :13
P. Aydogdu / 2011 / Chain conditions on non-summands / J. Algebra Appl. 10(3):475~489
A. Ghorbani / 2015 / Indecomposable decomposition and couniserial dimension / Bull. Math. Sci. 5(1):121~136
K. R. Goodearl / 1976 / Ring Theory / Marcel Dekker, Inc.
K. R. Goodearl / 2004 / An Introduction to Noncommutative Noether- ian Rings, second edition, London Mathematical Society Student Texts / Cambridge University Press :61
P. Aydogdu / 2011 / Chain conditions on non-summands / J. Algebra Appl. 10(3):475~489
A. Ghorbani / 2015 / Indecomposable decomposition and couniserial dimension / Bull. Math. Sci. 5(1):121~136
K. R. Goodearl / 1976 / Ring Theory / Marcel Dekker, Inc.
K. R. Goodearl / 2004 / An Introduction to Noncommutative Noether- ian Rings, second edition, London Mathematical Society Student Texts / Cambridge University Press :61
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