Decomposability of quotient polynomial ring I know module decomposability but I can't see this for a quotient polynomial ring. I know I need to look if there is an idempotent element or not but how  can I do that?

For example, let's look at $K[x]/\langle x^3-1\rangle$. $K$ is a field and $\operatorname{char} K \not = 3$. Does it have an idempotent element ?

In general, how can I decide whether a given quotient polynomial ring is decomposable or not?
I need some reference for this. Also, any comment will be helpful.


$k[X]/(f)$ is indecomposable iff $f=p^e$, where $p\in k[X]$ is irreducible and $e\ge 1$. 

If $f=p_1^{e_1}\cdots p_t^{e_t}$, then by CRT we have $k[X]/(f)\simeq k[X]/(p_1^{e_1})\times\cdots\times k[X]/(p_t^{e_t})$. On the other hand, $k[X]/(p^e)$ is a local ring and thus has only trivial idempotents.  
In your case, $$k[X]/(X^3-1)\simeq k[X]/(X-1)\times k[X]/(X^2+X+1)\simeq k\times k[X]/(X^2+X+1),$$ so this ring is decomposable.
