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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 23rd 2009

saw activity at simple object and started a tiny section with examples.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJul 29th 2019

Looking at this article: what is the need for having a zero object as opposed to just a terminal? In other words, would it be worse to say an object is simple if it has precisely two quotients, itself and $1$?

I’m particularly interested in the case of rings and commutative rings. One definition of simple ring is that it has precisely two two-sided ideals, and this condition is equivalent to the one I’m proposing. Similarly, a simple commutative ring under my proposal is just a field.

(It may be that the concept becomes interesting only under additional assumptions, such as Barr-exactness or something in that vein. That’s why I phrased my question as I did: would my proposal be any worse than the one given?)

• CommentRowNumber3.
• CommentAuthorAli Caglayan
• CommentTimeJul 31st 2019

The definition of simple ring on the nlab is $R$ is simple as a bimodule. This seems to be equivalent to your definition. As far as I can tell your definition should subsume that definition of simple ring. I think the terminology becomes problematic for Lie algebras however.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeJul 31st 2019

Yeah, for Lie algebras there are practical reasons for excluding the 1-dimensional case.

But I’m asking a more general theoretical question (insofar as there should be a general “theory” of simple objects in categories).