I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series ... )(adsbygoogle = window.adsbygoogle || []).push({});

In Chapter 2: Linear Algebras and Artinian Rings, on Page 61 we find a definition of a refinement of a chain and a definition of a composition series.

The relevant text on page 61 is as follows:

In the above text, Cohn indicates that a refinement of a chain (added links) is a composition series for a module [itex]M[/itex], but then goes on to to characterise a composition series for a module [itex]M[/itex] as a chain in which [itex]C_r = M[/itex] for some positive integer [itex]r[/itex], and for which [itex]C_i/C_{i-1}[/itex] is a simple module for each [itex]i[/itex].

So then, is Cohn saying that if a refinement is not possible, then it follows that [itex]C_r =M[/itex] for some [itex]r[/itex] and [itex]C_i/C_{i-1}[/itex] is a simple module for each [itex]i[/itex]? If so, why/how is this the case?

Peter

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# Composition Series of Modules ... Remarks by Cohn

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