Prime ideals are special subsets of a ring that generalize the notion of prime numbers. An ideal $I$ in a ring $R$ is called prime if whenever the product of two elements $a$ and $b$ in $R$ belongs to $I$, then at least one of the elements must be in $I$. This concept is crucial in understanding the structure of rings, particularly in the context of Artinian and Noetherian rings, as well as in complete rings where properties related to prime ideals help determine the ring's behavior.
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