ideals occur in several
algebraic structures. They play
a similar rôle to
normal subgroups in
groups in that they
allow the
concept of
quotient structures,
isomorphism theorems
etc.
A right ideal I of a ring R is a nonempty
subset of R
such that
- a+b in I for a,b in I
- I is closed under right multiplication. i.e. for each a in I and r
in R we have ar in I.
A left ideal is defined similarly, except that it has to be closed under
left multiplication.
A two-sided ideal (or just ideal) is a right and left ideal of a ring.
For commutative rings there is no difference between the
concepts of right,left and two-sided ideals.
Examples of ideals
If I and J are right ideals then so is
I+J which consists of all sums i+j with
i in I and j in J. If I
and J are ideals then so is IJ which consists
of all finite sums i1j1+...+ikjk, with ir in I and js in J. If S is a subset of R the right ideal it generates
is denoted by SR and consists of all finite sums i1j1+...+ikjk, with ir in S and js in R. Note that SR is a right ideal of R.
Finally, if Ij is a family of right ideals then
the sum of this family consists of all finite sums
ij1 +...+ ij1
with ijk in Ijk. The sum is again a right ideal.
If a is an element of R then the ideal it generates
RaR consists of all finite sums r1as1 + ... + rtast with ri
and si in R. This is an ideal.
If S is a subset of R then the ideal of R
it generates is the sum of the ideals RaR, with a
in S. We sometimes write (a1,...,an) for the ideal generated by
{a1,...,an}.