Let $I$ and $J$ be ideals in $R$. Is the set $K= \{ ab \ | \ a\in I, b\in J \}$ an ideal in R?

Question

I've just assumed that this is false, since the problem statements says to compare it to a previous problem where $\{ a+b \ | \ a\in I, b\in J \}$ is ideal.

However, by trial and error I can't find two ideals where this doesn't hold.

Is this false, and if so what's the counterexample? I haven't found one thus far.

score 4 · Accepted Answer · edited Mar 17 '16 at 23:40

4

Consider (x,y) and (z,w) in Z[x,y,z,w]. Then your set isn't closed under addition: it contains xz and yw but not xz+yw.

edited Mar 17 '16 at 23:40

user26857

53,190

answered Mar 17 '16 at 23:13

Daniel McLaury

25,340

1

this works if we take $(x,y)$ twice and consider $x^2+y^2$. – Asinomás Mar 17 '16 at 23:20
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Or $I=J=\langle 2,X\rangle$ and consider $4+X^2$. – hmakholm left over Monica Mar 17 '16 at 23:24

Let $I$ and $J$ be ideals in $R$. Is the set $K= \{ ab \ | \ a\in I, b\in J \}$ an ideal in R?

1 Answers1