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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeJun 28th 2024
   • PermaLink
   Author: zskoda
   Format: MarkdownItexStub with a definition. <a href="https://ncatlab.org/nlab/revision/Goldie+rank/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Goldie+rank">current</a>

   Stub with a definition.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 28th 2024
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   Author: Urs
   Format: MarkdownItexCould 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?

   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?

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeJun 28th 2024
   • PermaLink
   Author: zskoda
   Format: MarkdownItexSlight extension. <a href="https://ncatlab.org/nlab/revision/diff/Goldie+rank/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Goldie+rank/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Goldie+rank">current</a>

   Slight extension.

   diff, v3, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 28th 2024
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   Author: Urs
   Format: MarkdownItexIs it the maximum $n$ such that $\oplus_{i = 1}^n I_i \;\;\subset\;\; R$, for $I_i$ being non-trivial ideals of $R$?

   Is it the maximum $n$ such that $\oplus_{i = 1}^n I_i \;\;\subset\;\; R$, for $I_i$ being non-trivial ideals of $R$?

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeJun 28th 2024
   • (edited Jun 28th 2024)
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   Author: zskoda
   Format: MarkdownItexSorry, 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 29th 2024
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   Author: Urs
   Format: MarkdownItexThanks. I think that's equivalent to what I guessed. Will make a brief entry for "[[internal direct sum]]" to make this precise.

   Thanks. I think that’s equivalent to what I guessed. Will make a brief entry for “internal direct sum” to make this precise.