Let $R$ be a ring, $M$ a semisimple, left $R$-module. So, there are simple modules $M_i$ for $i\in I$ with $$M=\bigoplus_{i\in I} M_i$$ Is the length of $M$ equal to the cardinality of $I$?
2026-04-03 10:36:42.1775212602
What is the length of a semisimple module, if defined?
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I would interpret what you're saying as the composition length of $M$, which is usually only discussed in the finite case. Yes, if $I$ is finite, the composition length of this module is $|I|$.
Perhaps some people do look into composition lengths of different infinite cardinalities and distinguish: I see no reason not to. I see several references to "infinite composition length" but I did not run into any specializing the length to pay attention to the cardinality.