Is the Set of Products of Two Ideals Always an Ideal?

[gordon53]

Let I,J be ideals of a ring R. Show that the set of products of elements of I,J need not be an ideal (by counterexample - I have been trying to use a polynomial ring).

 

[matt grime]

And how has it gone? What examples have you tried, and where has it gone wrong?

 

[gordon53]

I think I may have solved it now, and would appreciate confirmation or corrections:

Let R[x,y] be the ring of polynomials with real coefficients. Let I be the ideal containing all elements of C with zero constant term. Let J be the ideal containing all elements of C with zero constant term and zero coefficients of x, y and xy.

Then I has multiples of x, y, xy, x^2, y^2, etc.
And J has multiples of x^2, y^2, x*y^2, y*x^2, etc.

Now IJ contains x*x^2 = x^3 and y*y^2 = y^3.

But x^3 + y^3 is factorised uniquely (since to irreducible factors) as (x + y)(x^2 - xy + y^2). Neither of these polynomials is in J, and therefore the sum is not an element of IJ. So IJ is not closed under addition.