semiprime ideal
Let be a ring. An ideal of is a semiprime ideal if it satisfies the following equivalent conditions:
(a) can be expressed as an intersection of prime ideals of ;
(b) if , and , then ;
(c) if is a two-sided ideal of and , then as well;
(d) if is a left ideal of and , then as well;
(e) if is a right ideal of and , then as well.
Here is the product of ideals .
The ring itself satisfies all of these conditions (including being expressed as an intersection of an empty family of prime ideals) and is thus semiprime.
A ring is said to be a semiprime ring if its zero ideal is a semiprime ideal.
Note that an ideal of is semiprime if and only if the quotient ring is a semiprime ring.
Title | semiprime ideal |
---|---|
Canonical name | SemiprimeIdeal |
Date of creation | 2013-03-22 12:01:23 |
Last modified on | 2013-03-22 12:01:23 |
Owner | antizeus (11) |
Last modified by | antizeus (11) |
Numerical id | 11 |
Author | antizeus (11) |
Entry type | Definition |
Classification | msc 16D25 |
Related topic | NSystem |
Defines | semiprime ring |
Defines | semiprime |