Can a non-RE language be reduced to an RE language? Let $L$ be recursively enumerable and $U$ be non-recursively-enumerable. Is it possible to reduce $U$ to $L$ recursively, $U\leq_R L$? Personally, I do not think this is possible. If we can reduce $U$ to $L$ by a Turing machine that always halt and we also have an Turing machine that compute $L$ (may not halt), then we connect these two parts, we will have a Turing machine to compute $U$. Then $U$ is also recursively enumerable.

I was asked to show some language $L$ is recursively enumerable first, which was OK. But afterwards, I was asked to show some language which is known not recursively-enumerable, $L_{diag}=\{w_i|M_i\text{ does not accept } w_i\}$ can be reduced to it. I.e. $L_{diag}\leq_R L$ which does not make sense to me.
As you say, no non-RE language&nbsp;$U$ can be reduced to any RE language&nbsp;$L$, because, then, we could recognize&nbsp;$U$ using the reduction and the recognition algorithm for&nbsp;$L$.

The language&nbsp;$L_{diag}$ that you mention is indeed non-RE: it's the complement of an RE set that isn't recursive.  There is no recursive reduction from $L_{diag}$ to&nbsp;$L$.