§ Radical ideals, nilpotents, and reduced rings
§ Radical Ideals
A radical ideal of a ring is an ideal such that
That is, if any power of is in , then the element
also gets "sucked into" .
§ Nilpotent elements
A nilpotent element of a ring is any element such that there exists
some power such that .
Note that every ideal of the ring contains . Hence, if an ideal
of a ring is known to be a radical ideal, then for any nilpotent ,
since , since is radical, .
That is, a radical ideal with always contain all nilpotents! It will
contain other elements as well, but it will contain nilpotents for sure.
§ Radicalization of an ideal
Given a ideal , it's radical idea .
That is, we add all the elements needs to have for it to become a radical.
Notice that the radicalization of the zero ideal will precisely contain
all nilpotents. that is, .
§ Reduced rings
A ring is a reduced ring if the only nilpotent in the ring is .
§ creating reduced rings (removing nilpotents) by quotienting radical ideals
Tto remove nilpotents of the ring , we can create . Since
is the ideal which contains all nilpotents, the quotient ring will contain
no nilpotents other than .
Similarly, quotienting by any larger radical ideal will remove all nilpotents
(and then some), leaving a reduced ring.
A ring modulo a radical ideal is reduced
§ Integral domains
a Ring is an integral domain if . That is,
the ring has no zero divisors.
§ Prime ideals
An ideal of a ring is a prime ideal if
. This generalizes
the notion of a prime number diving a composite: .
§ creating integral domains by quotenting prime ideals
Recall that every ideal contains a . Now, if an ideal is prime, and if
, then either or (by the definition of prime).
We create . We denote as the image of
in the quotient ring .
The intuition is that quotienting by a , since if ,
we are "forcing" that in the quotient ring , if , then either
or , since ,
A ring modulo a prime ideal is an integral domain.
I learnt of this explanation from this
excellent blog post by Stefano Ottolenghi.