Prime ideals are special subsets of a ring that have a key property related to the multiplication of elements. Specifically, an ideal I in a ring R is prime if whenever the product of two elements a and b from R is in I, at least one of those elements must also be in I. This concept plays an important role in understanding the structure of rings, particularly in the context of C*-algebras, as they help define certain algebraic properties and facilitate the analysis of representations.
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