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
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.