|
|
In every commutative ring the rigth ideals are also left ideals and bisided
ideals. So there is no point to talk about rigth ideal.
Suppose R is noetherian and there exists an ideal I which is not generated by a
finite number of elements.
You must find an infinite ascending chain to prove by contradiction. Pick the
ideal I1,generated by a single element( a principal ideal),say x1,element which
belongs to I. What can you say about I1 knowing that it is included in I? Can
you find other ideal I2 such that I1 is strictly included in I2 and I2 is
included in I?
Hint: There exists in I an element which does not belong to I1. Why ?
|
|