In this note we extend the notions of prime and semiprime ideals of a semigroup S to arbitrary S-acts and develop some of their basic properties. In particular, we characterize semigroups all of whose ideals are prime (semiprime).
We give a partial answer to the problem: Given a connected finite CW Hopf space X, determine sets P of prime numbers such that every multiplication on the P-localization XP of X is a P-localization of a multiplication on X. A complete answer is given when X is of rank at most 2 or a simple Lie group.
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