For two ideals $I$ and $J$ prove that $I(R/J)=(I+J)/J$
Solution 1:
Well, if you use the definition, it's obvious: an element of $I\cdot R/J$ is the ideal of $R/J$ generated by $I$, so it is the set of finite sums of the form: $$\Bigl\{\sum_k i_kr_k+J\bigm\vert i_k\in I, r_k\in R\Bigr\}.$$ Now, as $I$ is an ideal in $R$, the set of these finite sums u=is nothing but $I$ itself, whence the formula $$I\cdot R/J=(I+J)/J.$$ Note that, by the second isomorphism theorem, the latter quotient is isomorphic to $I/I\cap J$.