A proper ideal 
of a ring 
is called semiprime if, whenever 
for an ideal 
of 
and some positive integer, then 
. In other words, the quotient
ring 
is a semiprime ring.
If 
is a commutative ring, this is equivalent to requiring that 
coincides with its radical (and in this case 
is also called an ideal radical).
This means that, whenever a certain positive integer power 
of an element 
of 
belongs to 
, the element 
itself lies in 
. A prime ideal is certainly
semiprime, but the latter is a strictly more general notion. The ideal 
of the ring of integers 
is not prime, but it is semiprime, since for all integers
, 
is a multiple of 
iff 
is, since both 2 and 3 must appear in its prime factorization.
The same argument shows that the ideal 
of 
is always semiprime if 
is squarefree. This is not necessarily the case when 
is a semiprime number, which causes a
conflict in terminology.
In general, the semiprime ideals of a principal ideal domain are the proper ideals whose generator has no multiple prime factors.
