Is there an infinite family $\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.

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

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

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.

Is there an infinite family $\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.