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Could you clarify “the maximum number of direct summands of left (or right) ideals” of a ring? Some qualification seems to be missing.
I guess you mean non-zero ideals? And you mean a direct sum of them that is still a sub-module of the ring itself?
Is it the maximum n such that ⊕ni=1Ii⊂R, for Ii being non-trivial ideals of R?
Sorry, I was not around. Thanks for the question.
The direct sum is the internal direct sum, hence a submodule by definition.
To quote Goodearl, the Goldie rank is n (the ring) “contains a direct sum of n nonzero submodules but no direct sum of n+1 nonzero submodules.” The definition is also used for modules, not only for the ring. Noetherianess is clearly sufficient to have a finite number.
For submodules to use this definition one assumes a finite rank, that is its injective envelope is a direct sum of a finite direct sum of indecomposables.
Thanks. I think that’s equivalent to what I guessed. Will make a brief entry for “internal direct sum” to make this precise.
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