## Direct product of rings

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.

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 I have no time right now, but if I did have time the first place I'd look is T.Y. Lam's Lectures on Modules and Rings. I think there are a number of pages available through Google Books. The answer might be there – David White Dec 29 at 19:37 @David White: Exactly, its always true that there are a number of pages available through Google Books ! – chatish Dec 29 at 21:33