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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.

edited Jan 9 at 14:26

asked Dec 29 at 7:52

chatish
2,193●4●20

	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

Fred Rohrer · Answer 1 · 2012-12-29 22:15:03Z UTC

4

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.)

edited Dec 29 at 22:15

answered Dec 29 at 21:30

Fred Rohrer
1,521●6●11

@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. – chatish Dec 29 at 21:52

1

But that von Neumann regular rings are semi-hereditary seems not to be in doubt, so that answers one question. – Todd Trimble Dec 29 at 21:54

Oh dear, let me edit my silly mistake. – Fred Rohrer Dec 29 at 21:57

Direct product of rings

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