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. 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 n Z {displaystyle nmathbb {Z} } where n is a square-free integer. So, 30 Z {displaystyle 30mathbb {Z} } is a semiprime ideal of the integers (because 30 = 2 × 3 × 5, with no repeated prime factors), but 12 Z {displaystyle 12mathbb {Z} ,} is not (because 12 = 22 × 3, with a repeated prime factor). The class of semiprime rings includes semiprimitive rings, prime rings and reduced rings. Most definitions and assertions in this article appear in (Lam 1999) and (Lam 2001). For a commutative ring R, a proper ideal A is a semiprime ideal if A satisfies either of the following equivalent conditions: