Direct product of rings - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T06:45:40Z http://mathoverflow.net/feeds/question/117486 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117486/direct-product-of-rings Direct product of rings chatish 2012-12-29T07:52:19Z 2013-01-09T14:26:47Z <p>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 ?</p> <p>I think the answer must be negative but i have no proof or counterexample yet. </p> http://mathoverflow.net/questions/117486/direct-product-of-rings/117564#117564 Answer by Fred Rohrer for Direct product of rings Fred Rohrer 2012-12-29T21:30:55Z 2012-12-29T22:15:03Z <p>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 <em>Lectures on modules and rings,</em> Example 2.32 d).</p> <p>(In the above, fields and rings are not necessarily commutative.)</p>