Euclidean Function at 0 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:59:18Z http://mathoverflow.net/feeds/question/31681 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31681/euclidean-function-at-0 Euclidean Function at 0 Yiftach Barnea 2010-07-13T10:49:58Z 2010-07-13T14:09:43Z <p>In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for non-zero elements. I am teaching an undergraduate course and I am trying to make things simpler as possible. Is there any good reason why not to define it as \$0\$ at \$0\$? </p> http://mathoverflow.net/questions/31681/euclidean-function-at-0/31686#31686 Answer by BS for Euclidean Function at 0 BS 2010-07-13T11:46:18Z 2010-07-13T12:17:49Z <p>I think this is only because you don't care dividing by zero. But you may as well define the value at zero to be 0, and any other value at least 1. See <a href="http://www.math.uconn.edu/~kconrad/blurbs/ringtheory/euclideanrk.pdf" rel="nofollow">this</a> expository paper by Keith Conrad for interesting remarks on euclidean domains and functions.</p> http://mathoverflow.net/questions/31681/euclidean-function-at-0/31704#31704 Answer by Bill Dubuque for Euclidean Function at 0 Bill Dubuque 2010-07-13T14:03:57Z 2010-07-13T14:09:43Z <p>You'll find your answer and much more in the little-known paper [1] which surveys all of the dozen known ways of axiomatizing Euclidean rings (including those of Nagata and Samuel), and explores in-depth all of their logical interrelations. It's a convenient reference to have at hand when you're comparing texts which use (seemingly) different definitions of Euclidean rings / domains.</p> <p>[1] Euclidean Rings. A. G. Agargun, C. R. Fletcher<br> Tr. J. of Mathematics, 19, 1995, 291 - 299.<br> <a href="http://journals.tubitak.gov.tr/math/issues/mat-95-19-3/pp-291-299.pdf" rel="nofollow">http://journals.tubitak.gov.tr/math/issues/mat-95-19-3/pp-291-299.pdf</a> </p>