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 $\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. 


Barbara Osofsky proves this is not possible (assuming the Axiom of Choice, I think) in "Noninjective cyclic modules", Proc. Amer. Math. Soc. 19 (1968), 13831384. 


Products of fields are semihereditary. This follows from the facts that products of fields are von Neumann regular and that von Neumann regular rings are semihereditary. A proof can be found (as suggested by David White) in Lam's Lectures on modules and rings, Example 2.32 d). (In the above, fields and rings are not necessarily commutative.) 


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), 149165, 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. 

