Main Index
Algebraic structures
Ring theory
Basic notions
Subject Index
comment on the page
Main Index
Algebraic structures
Ring theory
Basic notions
Subject Index
comment on the page
A proper ideal 
of 
is called radical of an ideal 
of 
, if 
for some positive integer 
implies that 
, i.e.
The name goes back to the fact that the radical of an ideal 
is the set of all the possible “roots” of elements of 
.
The radical of an ideal 
of 
, can be defined also as follows: Consider the projection 
. Then the radical 
is the preimage of the nilradical 
↑ of 
.
Theorem. We have
if nad only if 
is a prime ideal then 
for every positive integer 
.Theorem. The radical of an ideal 
is the intersection of all prime ideals containing 
.
Theorem. If the radical 
, 
of ideals 
of a ring 
are coprime ↑, then also the ideals 
are coprime.
Cite this web-page as:
Štefan Porubský: Radical.