A and B left ideals of ring R. Is $BA⊆A$?

Question

Let $R$ be a ring. Let $A$ be left ideal of $R$, and $B$ be a left ideal of $R$.

Is there any way I could show that $BA⊆A$?

I was trying to use this fact to help me with another question, but I'm not convinced it is true.

Thanks

Answer 1

$BA \subseteq RA \subseteq A$.

The second containment is because $A$ is a left ideal, the first because $B \subseteq R$ (we don't need $B$ to be a left ideal for this).

Answer 2

As user73985 has already pointed out, this follows trivially from $BA\subseteq RA\subseteq A$, and $B$ could be any nonempty subset of $R$ in fact.

Showing "$AB\subseteq A$" fails is slightly more interesting.

If $F$ is a field and $A$ is the left ideal of $M_2(F)$ made up of matrices with zeros in the right column, and $B$ is the left ideal made up of matrices with zeros in the left column, then $AB\subseteq B$, but $A\cap B=\{0\}$, so certainly $AB\nsubseteq A$.