Semiprime ring

In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical ideals.

For example, in the ring of integers, the semiprime ideals are the zero ideal, along with those ideals of the form ${\displaystyle n\mathbb {Z} }$ where n is a square-free integer. So, ${\displaystyle 30\mathbb {Z} }$ is a semiprime ideal of the integers (because 30 = 2 × 3 × 5, with no repeated prime factors), but ${\displaystyle 12\mathbb {Z} \,}$ is not (because 12 = 2² × 3, with a repeated prime factor).

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