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Concept
Semiprime ring
Summary
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 and semiprime rings are the same as reduced rings.
For example, in the ring of integers, the semiprime ideals are the zero ideal, along with those ideals of the form where n is a square-free integer. So, is a semiprime ideal of the integers (because 30 = 2 × 3 × 5, with no repeated prime factors), but 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 and .
For a commutative ring R, a proper ideal A is a semiprime ideal if A satisfies either of the following equivalent conditions:
If xk is in A for some positive integer k and element x of R, then x is in A.
If y is in R but not in A, all positive integer powers of y are not in A.
The latter condition that the complement is "closed under powers" is analogous to the fact that complements of prime ideals are closed under multiplication.
As with prime ideals, this is extended to noncommutative rings "ideal-wise". The following conditions are equivalent definitions for a semiprime ideal A in a ring R:
For any ideal J of R, if Jk⊆A for a positive natural number k, then J⊆A.
For any right ideal J of R, if Jk⊆A for a positive natural number k, then J⊆A.
For any left ideal J of R, if Jk⊆A for a positive natural number k, then J⊆A.
For any x in R, if xRx⊆A, then x is in A.
Here again, there is a noncommutative analogue of prime ideals as complements of m-systems. A nonempty subset S of a ring R is called an n-system if for any s in S, there exists an r in R such that srs is in S. With this notion, an additional equivalent point may be added to the above list:
R\A is an n-system.
The ring R is called a semiprime ring if the zero ideal is a semiprime ideal. In the commutative case, this is equivalent to R being a reduced ring, since R has no nonzero nilpotent elements.
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