2 Edited Title

Is there an infinite family $(R_\alpha) \lbrace R_\alpha\rbrace_\alpha$ of rings (with identity $1\neq 0$) such that their direct product is a (semi) hereditary ring ?

I think the answer must be negative but i have no proof or counterexample yet.

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# Direct product of rings

Is there an infinite family $(R_\alpha)$ of rings (with identity $1\neq 0$) such that their direct product is a (semi) hereditary ring ?

I think the answer must be negative but i have no proof or counterexample yet.