# Homework Help: Ring axioms

1. Sep 11, 2010

### dndod1

1. The problem statement, all variables and given/known data
This is not an assignment question, just something that I am wondering about as an offshoot of an assignment question.
In my course notes Rings are defined as having 3 axioms and commutative rings have 4.(outined below)
I have just answered this question:
Show that the axiom set (sigma) is not independent for the following axioms.
B1 (S,+,*) is right distributive
B2 (S,+,*) is left distributive
B3 (S, *) is commutative

So I used B1 and B3 to derive B2. Therefore the axioms are not independent.

What I am wondering is: This appears to be trying to make me think about the difference between rings and commutative rings. So this means that the axioms for a commutative ring are not independent. Are the axioms for a ring independent? Or does it depend on the wording? (ie whether you group the 2 distributive laws as a single axiom?)
It appears to me that the wording of axioms is not universal; is that correct? (Some books seem to include closure as an axiom, others don't.)

2. Relevant equations
I have as my ring axioms
R1 The additive structure is an abelian group
R2 The multiplicative structure is a semigroup
R3 Two distributive laws connect the additive and multiplicative structures.

For commutative ring add R4 The multiplicative structure is commutative.

b]3. The attempt at a solution[/b]

If I gave axioms as R1-4 above, they would be independent?????
If I gave them as listed below they wouldn't be independent???? Is this correct?

R1 The additive structure is an abelian group
R2 The multiplicative structure is a semigroup
R3 LEFT distributive law connects the additive and multiplicative structures.
R4 RIGHT distributive law connects the additive and multiplicative structures.

For commutative ring add R5 The multiplicative structure is commutative.

It would be nice to be clear in my head on this idea!
Many thanks is anticipation.