# 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 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 '12 at 19:37
@David White: Exactly, its always true that there are a number of pages available through Google Books ! –  user30230 Dec 29 '12 at 21:33

Barbara Osofsky proves this is not possible (assuming the Axiom of Choice, I think) in "Noninjective cyclic modules", Proc. Amer. Math. Soc. 19 (1968), 1383-1384.

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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.)

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@Fred Rohrer: But an infinite direct product of non-zero rings can not be countable so there is a gap in your argument for the hereditary case. –  user30230 Dec 29 '12 at 21:52
But that von Neumann regular rings are semi-hereditary seems not to be in doubt, so that answers one question. –  Todd Trimble Dec 29 '12 at 21:54
Oh dear, let me edit my silly mistake. –  Fred Rohrer Dec 29 '12 at 21:57