As to the question of "why?", I have a very biased and unscientific answer: Sup is the most awesome category.
Oh, *there*'s the problem. I was getting quite puzzled about all this stuff. Presumably by Sup you mean what Peter Johnstone calls CSLat, complete semilattices, which is a lovely self-dual category. (If not ignore the following.) According it the status of "the most awesome" however is a symptom of not yet having come to grips with the joy of Chu, a more awesome self-dual category (fully) embedding CSLat in a duality-preserving and concrete-preserving way while exhibiting that duality as simply matrix transposition, yet still not *the* most awesome. And all that just in Chu(Set,2). Chu(Set,8) embeds Grp, and concretely at that, which is more awesome but still not awesome to the max. More awesome yet is that you can concretely embed every category of relational structures of total arity n in Chu(Set,2^n)---Grp fits that description on account of the group multiplication being a ternary relation, whence Chu(Set,8)). And so on. If going up only reduces the awe, then one should instead go down from CSLat for greater awe. God and the devil command a degree of awe that the middle class is hard pressed to match. Not only am I not a ring theorist but it's never occurred to me even to play one on the Internet. On the matter of the ideals of R, it would be very nice if they were just the endomorphisms of R but presumably that doesn't work on the ground that not every quotient of R embeds as a subring of R---if that's wrong then I'm really confused. I'm not a category theorist either but I do try. Isn't the obvious gadget to extract from R not its lattice of ideals but its category of quotients suitably defined? Bill, is that what you were getting at? Vaughan