Understanding the definition of $UV$, where $U$ and $V$ are ideals in a ring

Question

I have the following question at hand: I. N. Herstein Topics in Algebra: Ideals and Quotient Rings : Qn $3.4.6$

If$\ \ U,V$ are ideals of $\ R\ $,let $UV$ be the set of all elements that can be written as finite sums of elements of the form $uv$ where $u\in U$ and $v\in V$. Prove that $UV$ is ideal in $R$.

I am a beginner and I am having a problem in understanding the meaning of the question especially, the "all elements written as finite sums of elements of the form $uv$". Can someone explain to me (preferably by an example)?

Answer 1

In a set notation we have $UV=\{\sum_{i=1}^n u_iv_i : n\in \mathbb{N}, u_i\in U, v_i\in V\}$

For two elements in $UV$ the number of terms of the form $u_iv_i$ that we sum do not need to be the same.

Here is an example. Let $U=\langle x+3\rangle$ and $V=\langle x,2\rangle$ are ideals in $\mathbb{R}[x]$.

The following are examples of elements in $UV$

1) $(x+3)\cdot 4+ (x+5)(x+3)\cdot x$

2) $(x+3)\cdot (x+4) $