Which of the following is also an ideal?

Question

If $U,V$ are ideals of a ring $R$, then which of the following is also an ideal of $R$?

1. $U+V=\{u+v\mid u\in U,v\in V\}$

2. $U\cdot V=\{u\cdot v\mid u\in U,v\in V\}$

3. $U\cap V$

My attempt: I have proved $U+V$ and $U\cap V$ are ideals of $R$.

My problem: I have problem about the second option. I think it is not ideal of $R$. If I am wrong then please prove this otherwise give one counter example to show that $U\cdot V$ is not ideal of $R$.

Answer 1

In the ring $\mathbb{Z}[X,Y]$, what happens if we take $U=V=(X,Y)$? Does $X^2 + Y^2$ lie in $U.V$?

Answer 2

If you define $U.V = \{uv : u \in U, v \in V\}$, the resulting set is not necessarily an ideal as it is not necessarily closed under addition. For instance, consider the example given by Slade.

This is why the multiplication of ideals $U,V$ is usually defined as $U.V = \{u_1v_1 + \cdots + u_nv_n : u_i \in U, v_i \in V, n \in \mathbb{N}\}$.