If you are in a PID (principal ideal domain) and there is an element p in the domain that is irreducible, how can you show that
(the principal ideal generated by p) is maximal?
I know in a PID principal means prime, and there is an ascending chain order, but I don't really know how to bring it all together.
Thanks!
2007-03-08 14:37:43 · 1 answers · asked by mobaxus 2 in Science & Mathematics ➔ Mathematics
I think I'm remembering all the definitions correctly, I haven't thought about PIDs and such for a while. Anyhow, suppose there is some other ideal containing
. Then since you're in a PID this ideal is generated by a single element, say q. Thus, since
is contained in . Thus the only ideal properly containing is the whole ring, which means is maximal.
2007-03-08 14:51:58
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answer #1
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answered by Sean H 5
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, there is an element of the ring such that p = r*q. Since p is irreducible, either r or q must be a unit. If q is a unit, then
is the whole ring. If r is a unit, then q = r^{-1}*p, and so
is contained in