Fork me on GitHub
Math for the people, by the people.

User login

ring

Defines: 
multiplicative identity, multiplicative inverse, ring with unity, unit, ring addition, ring multiplication, ring sum, ring product, unital ring, unitary ring
Type of Math Object: 
Definition
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

16-00 no label found20-00 no label found13-00 no label found81P10 no label found81P05 no label found81P99 no label found

Comments

if it were me I'd get rid of the sentence about commutative rings. I'd probably also express it in terms of additive abelian groups and multiplicative semigroups or monoids, so maybe you shouldn't listen to me.

I think he's planning on still having a separate "commutative ring" entry...
-apk

Can't a ring (R,+,*) be described as a commutative group (R,+) and a * (closed) operation?

yes, if you also ask
for * being associative
and add a distributive law that links + with *

f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

This definition is not adequate because we are missing the very important property of distribution. Distribution ensures other structural features of the ring, specifically with the zero.

In Dummit & Foote it is proved that the following definition of a ring "A ring (R,+, ·) is a group (R, +) with an additional associative multiplication · with identity, which distributes over the addition." implies commutativity in the additive group. Yet authors persist to include commutativity as an axiom in Ring Theory.

Any comments?

The requirement that a ring have a multiplicative identity is not, in general, a standard ring axiom. I suppose the reason that authors explicitly list the requirement that $(R,+)$ be abelian is that it may not be immediately obvious that commutativity of $+$ is implied by the left and right distributive laws. It wasn't to me anyway :)

Keenan

It's really just a matter of taste. Commutativity of addition isn't just a nice property that unital rings happen to have, it's part of what being a ring is all about. We want it even for non-unital rings (where it isn't automatic). So it's natural to include it in the definition.

Note that djao's definition doesn't include a multiplicative identity, so he has no choice but to specify commutativity of addition explicitly. So your question appears to have no relation to the title of your post.

You are right about the relation with djao's post. ( I am a bit tired. ) - Maybe it's all caused by the 1, do we include a 1 or not? I noticed there are a lot of competing definitions. A matter of taste, as you say.

Thanks for your reply.

Could you add two entries:

1. A proof that multiplicative identity and distributive
law imply commutativity of addition.

2. An non-trivial example of an algebraic system where
multiplication distributes over addition but addition is not
commutative. (Any noncommutive group is a trivial example
if we define the product of any two elements to be zero.)

Ok, after work.

rspuzio wrote:
> 1. A proof that multiplicative identity and distributive
> law imply commutativity of addition.

By the way, this is exercise 2.1.3 in Jacobson's
_Basic Algebra I_.

> 2. An non-trivial example of an algebraic system where
> multiplication distributes over addition but addition is not
> commutative. (Any noncommutive group is a trivial example
> if we define the product of any two elements to be zero.)

Jacobson gives a construction in section 2.17 to embed a
``rng'' (what I would call a ring without unity) into a ring
with unity. Following his idea, let (R, +, .) be a tuple
for which (R, +) is a group, (R, .) is a semigroup, and .
distributes over + on the left *and* on the right. Define a
tuple (S, +, .) by S = Z \times R, where (S, +) is the
direct product of (Z, +) and (R, +), and the semigroup
structure (S, .) is given by the multiplication

(m, a)(n, b) = (mn, mb + na + ab).

It is not difficult to verify that (S, .) is a monoid with
unit (1, 0) and that . distributes over +. By exercise
2.1.3, it follows that (S, +) is abelian. Moreover, there
is an *injective* homomorphism f : R -> S defined by
f(a) = (0, a), so we can pull the abelian property back to R.

(I would *really* appreciate it if anyone could detect and
point out errors in the above reasoning.)

This really goes back to Taussky-Todd's 1936 paper ``Rings
with non-commutative addition''. Taussky-Todd proves that
in any distributive near-ring (N, +, .), elements of
NN = { ab : a, b in N } commute additively with each other.
As a consequence, if N is a distributive near-ring and
NN = N, then N is actually a ring.

To get a nontrivial example where commutativity fails, you
may have to be satisfied with permitting distributivity only
on *one* side. You might look into endomorphism near-rings.
I found the books _Near-Rings_ (by Pilz) and _Near-rings and
their links with groups_ (by Meldrum) helpful in composing
this post, but I don't know enough about this area to give
you a concrete example I know is correct.

The person who contributed PM's entry on near-rings,
J\"urgen Ecker, is a current researcher in the field and is
one of the author's of GAP's SONATA package for working with
near-rings, but he no longer logs in to PM.

> author's

This is why PM needs post-editing capabilities.

>
> It is not difficult to verify that (S, .) is a monoid with
> unit (1, 0) and that . distributes over +. By exercise
> 2.1.3, it follows that (S, +) is abelian. Moreover, there
> is an *injective* homomorphism f : R -> S defined by
> f(a) = (0, a), so we can pull the abelian property back to
> R.
>
> (I would *really* appreciate it if anyone could detect and
> point out errors in the above reasoning.)
>

What errors are you speaking of?

> What errors are you speaking of?

I meant ``Which claim did I make that was incorrect?''. (We
know that there must be an error somewhere, since it directly
contradict's rspuzio's example: *any* group (G, +), even a
nonabelian one, satisfies the assumptions I gave if (G, .)
has the trivial product.)

As it turns out, the incorrect claim is ``(S, .) is a monoid''.
It is easy to show that with the defined multiplication,
(S, .) is a unital magma. However, proving that . is
associative requires that (S, +) be abelian. In my
attempted proof (not posted) I implicitly assumed (S, +)
was abelian even though it wasn't supposed to be.

A lesson is that one should not rely too much on notation.

Or, maybe that ab=0 for all a, b will render S=0, if S contains a multiplicative unity 1, since a=1.a=0.

Subscribe to Comments for "ring"