A quotient of infinite direct product of field Let $R$ be a ring of countable infinite direct product of a field $F$ and $I$ be the countable infinite direct sum of $F$,clearly $I$ is an essential ideal of $R$.
Is $R$/$I$ an indecomposable $R$-module?
Is $R$/$I$ a cyclic $R$-module?
Is $I$ a maximal ideal in $R$?
I will be thankful for any clarification.

As it is a countable product we will asume that it is indexed in the natural. 
1) We will put $e\in R$ such that $e_i=1_F$ if $i=2k$ for some $k\in\mathbb{N}$ and $0_F$ otherwise, and $f\in R$ such that $f_i=1_F$ if $i=2k+1$ for some $k\in\mathbb{N}$ and $0_F$ otherwise. It is clear that $1_R=e+f$, $e$ and $f$ are central idempotents orthogonals. Then $R=Re\oplus Rf$. Moreover $e+I$ and $f+I$ are central idempotents orthogonals non zero and $1_R+I=(e+I)+(f+I)$. Therefore $R/I=(R/I)(e+I)\oplus(R/I)(f+I)$.
2)As mentioned in the comments $R/I=R(1+I)$.
3) It is not maximal as $R/I$ is not a field, not even a integer domain $(e+I)(f+I)=0$.
