Htykuut

¡Hello! Please give me a hint for this problem.

" Let $R$ a ring, $L$ a left ideal of $R$, $I$ right ideal of $R$. Show if $M$ are a left $R$-module, then exists a bijective function of commutative groups.

$$ f colon (R/I) otimes_{R} M to M/IM$$ such that

$ f((r+I) otimes m) = rm + IM$. And show that as commutative groups

$$ (R/I) otimes _{R} (R/L) cong R/(I+L)$$

If we tensor with $M$ the short exact sequence $0to Ito Rto R/Ito 0$ and consider the exact sequence $0to IMto Mto M/IMto 0$, we obtain a commutative diagram of abelian groups with exact rows
$$require{AMScd}
begin{CD}
{} @. Iotimes_RM @>>> Rotimes_RM @>>> (R/I)otimes_R M @>>> 0 \
@. @VaVV @VbVV @VcVV @. \
0 @>>> IM @>>> M @>>> M/IM @>>> 0
end{CD}
$$
We have $a(rotimes x)=rx$, $b(rotimes x)=rx$ and $c((r+I)otimes x)=rx+IM$. Note that $a$ is surjective and $b$ is an isomorphism. A very simple diagram chasing shows that $c$ is an isomorphism as well.

For the second case, we need to compute
$$
I(R/L)=(IR+L)/L=(I+L)/L
$$
and so
$$
(R/L)big/I(R/L)=(R/L)big/((I+L)/L)cong R/(I+L)
$$
by the homomorphism theorem.