Let $k$ be a field and $A$ a finitely generated algebra over $k$. I know that if $A$ is finite dimensional as a vector space over $k$, then $A$ can be decomposed as a product of indecomposable $k$-algebras: $$A=A_1\times\ldots\times A_n$$ My question is if there is a similar decomposition of an arbitrary $k$-algebra.

I'm not sure how to approach this problem because of the existence of descending chains, for example in $k[x]$ there is a chain: $$\langle x\rangle\supseteq\langle x^2\rangle\supseteq\langle x^3\rangle\supseteq\ldots$$ and the existence of idempotents with arbitrary degrees. I also tried to show the existence of an $N\in\mathbb{N}$ such that every idempotent in $k[x_1\ldots,x_n]/\langle f_1,\ldots,f_m\rangle$ can be seen as a sum of idempotents in $k[x_1\ldots,x_n]/\langle f_1,\ldots,f_m,x_1^N,\ldots,x_n^N\rangle$. But I have not had luck, so any help would be appreciated.

Thanks, Luis

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