### Updated images of mugs

In case anyone still has an opinion... which looks cooler? I am asking because no one seems to be answering! (Well it doesn't really concern anyone, but it is because I am stuck in the middle about which to choose.) The cost and everything as been sorted, we just need to pick one of the two... (by the way, the text is on the wrong side in the images but that can be changed).

Oh, yeah, I am actually revising algebra and ideals but because they stink really really badly I was drawn to this. Seriously--I hate cyclic subgroups and ideals seem to be generated by them and that leaves me perplexed. A cyclic subgroup is \lange a \rangle = {a^n | n is an integer}. Now we interpret a^n= a*a*..... *a and star is whatever binary operation the group has.

Yes, I'm probably sounding very dumb, but in rings since we have the group (R, +) do we take the bin. op in the ideal generated by a to be addition? But then GAH. Yes--this was the hole I was talking about. Ideals require that for a \in I (the ideal) and r \in R (the ring) ar and ra \in I. So ....

Something is not making sense. Did I mention that I don't like algebra that much any more?

## 1 comment:

Doh! Yes--I have banged my head on the wall, and you might want to bang my head too! (Big cheers for Prof. P for his prompt reply to my stupid email..) Parts of the email reply are below:

"The l/rangle notation is being used in two different contexts, with different meanings in each context.

In the context of groups it means the cyclic subgroup generated

by a (so all a^n).

In the context of rings it means the ideal generated by a (so all

expressions of the form r_1as_1+...+r_nas_n) - so not relating to a single binary operation in this case.

...in general regard this as a single notation for two

completely different things (and which applies depends on the

context - groups or rings)."

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