On Wed, Mar 18, 2009 at 7:17 AM, Kalman Noel <noel.kalman@googlemail.com> 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.