So a clearer reframing might be: “Ring is like Field, but without multiplicative inverse”.
On Wed, Mar 18, 2009 at 7:17 AM, Kalman Noel <[email protected]>wrote: > Wolfgang Jeltsch schrieb: > > Okay. Well, a monoid with many objects isn’t a monoid anymore since a > monoid > > has only one object. It’s the same as with: “A ring is a field whose > > multiplication has no inverse.” One usually knows what is meant with this > but > > it’s actually wrong. Wrong for two reasons: First, because the > multiplication > > of a field has an inverse. Second, because the multiplication of a ring > is > > not forced to have no inverse but may have one. > > “A ring is like a field, but without a multiplicative inverse” is, in my > eyes, an acceptable formulation. We just have to agree that “without” > here refers to the definition, rather than to the definitum. > >
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