[E-Lang] ERTP-aware MintMaker

Ben Laurie ben@algroup.co.uk
Thu, 15 Feb 2001 16:50:32 +0000


"Jonathan S. Shapiro" wrote:
> 
> Ben Laurie wrote:
> > Ahem. Speaking as a defrocked mathematician, group arithmetic is _not_
> > modular arithmetic.
> 
> Okay. Adding (briefly) to the digression, I suspect that I have dropped
> some bits that have led me to a broken mental association between groups
> and ring algebras. Can you straighten me out on this?

Hmm. Well, really briefly, a group is a set that has an operation (on an
ordered pair of elements) under which the group is closed, each element
has an inverse and there is an identity element. This includes things
like addition over all integers, modulo addition over cyclic groups
(err, ok, that was a little circular), multiplication over integers (but
_not_ division), rotations and reflection of regular polygons (the
dihedral groups, IIRC) and so on.

A ring is a group plus a second operation (one usually thinks of the
second operation as "multiplication" and the first as "addition") where
the second operation is nearly a group operation, but does not have an
identity element.

A ring with a one has the identity element for the second operation (the
"one" coz it is "multiplication" - the first identity being thought of
as "zero").

Unless you come from Cambridge (as I do), in which case a ring is called
a rng and a ring with a one is called a ring. Hilarious, eh?

Of course, modulo arithmetic on integers is a ring with a one (actually,
its something with even more properties than those, but let's stop
there, eh?).

Did that help at all?

Cheers,

Ben.

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