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. 



Direct product of rings
As @Jeremy Rickard has mentioned, the impossibility was shown by Osofsky in 1968. But it is interesting that there is another article dating back to 1968 (again!) that proves the same thing. See **Cateforis, Sandomerski, The singular submodule splits off, J. Algebra, 10 (1968), 149-165, Theorem 4.1.** This article was received by J. Algebra on **November 20, 1967** while Osofsky's paper was received by Proc. Amer. Math. Soc. on **July 10, 1967**. I hope this information meets the expectations. 


