Let $K$ be a field and let $I$ be an infinite set. We put $R = K^I$ (copies of $K$), $J = K^{(I)}$ and $S = \{1 − r \mid r ∈ J\}$. Then $R/J\simeq S^{−1}R$, $R$ is an injective module, but $R/J$ is not injective.
My questions are:
1) $R$ is injective as $K-$module or $R-$module?2) $R/J$ is not injective as $K-$module or $R-$module?
Thanks.