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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.
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