Rings such that every quotient has an indecomposable decomposition Let $R$ be a commutative ring with identity. We say that $R$ has an indecomposable decomposition if it can be written as a finite direct sum of indecomposable rings.
Is there any characterization for a ring $R$ such that its every homomorphic image has an 
indecomposable decomposition?
 A: Here is a characterization, though I don't know if it's any more useful than the definition itself.  A ring $R$ has your property iff whenever $A$ is an infinite set of maximal ideals of $R$, there is some $M\in A$ such that $\bigcap_{N\in A,N\neq M} N\subset M$.
To prove this, suppose that there exists an infinite set of maximal ideals $A$ such for every $M\in A$, $\bigcap_{N\in A,N\neq M} N\not\subset M$.  Let $I=\bigcap_{M\in A} M$ and consider the quotient $R/I$.  For every $M\in A$, $\{M\}$ is clopen in is $\operatorname{Spec} R/I$:  it is closed since $M$ is a maximal ideal, and open since if $f\in \bigcap_{N\in A,N\neq M} N\setminus M$ then $M$ is the only prime of $R/I$ not containing $f$.  Thus $\operatorname{Spec} R/I$ has infinitely many clopen sets and $R/I$ contains infinitely many idempotents.
Conversely, suppose there is an ideal $I$ such that $R/I$ is not a finite product of indecomposable rings.  By König's lemma we can then find an infinite sequence $\{e_n\}$ of distinct idempotents in $R/I$ such that $e_ne_{n+1}=e_{n+1}$.  Setting $f_n=e_n-e_{n+1}$, the $f_n$ are an infinite set of nonzero orthogonal idempotents.  For each $n$, let $M_n$ be a maximal ideal such that $1-f_n\in M_n$.  Then $A=\{M_n\}$ is an infinite set of maximal ideals such for every $M\in A$, $\bigcap_{N\in A,N\neq M} N\not\subset M$ (namely, if $M=M_n$, then $f_n\in \bigcap_{N\in A,N\neq M} N\setminus M$).
