Question

Let I and J be ideals in R. Is the set $\mathit{K}\mathbf{=}\mathbf{\{}\mathbf{ab}\mathbf{/}\mathbf{a}\mathbf{\in }\mathbf{I}\mathbf{,}\mathbf{b}\mathbf{\in }\mathbf{J}\mathbf{\}}$an ideal in R? Compare Exercise 20.

Short answer

It is proved that K is not an ideal in R.

Step-by-step solution

Given statement

It is given that I and J are assumed to be ideals in R. But in general, the set is not ideal.

Show that K is ideal or not in R

Assume that $R=\mathrm{\mathbb{Z}}\left[x\right],I=\left(2,x\right)$and$J=\left(3,x\right)$ , so that I and J are the ideals of polynomials with constant multiples of 2 and 3 respectively.

Since,$3x-2\in IJ$, but $x\notin IJ$, then $IJ$is not closed under addition.