Explore Dependence of Axioms for Rings & Commutative Rings

[dndod1]

Homework Statement

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.)

Homework 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.

 

[nate808]

it is important to have clear and precise definitions and axioms in any field of study. In the case of rings and commutative rings, the axioms may vary slightly depending on the source, but the core principles remain the same. In general, the axioms for a ring are considered independent, as each one is necessary to fully define the structure. However, as you have observed, the axioms for a commutative ring are not independent, as the commutative property can be derived from the other axioms.

In terms of the wording of the axioms, it is important to be consistent and clear in how they are defined. Some sources may include closure as an axiom, while others may consider it a consequence of the other axioms. Ultimately, the important thing is that the axioms accurately and fully describe the structure being studied.

To answer your specific questions, if you listed the axioms for a ring as R1-4, they would still be considered independent. However, if you listed them with the distributive laws as separate axioms (R3 and R4), then they would not be independent, as the distributive laws can be derived from the other axioms.

I hope this helps clarify the concept of independent axioms in the context of rings and commutative rings. It is always important to carefully consider and define the axioms in any mathematical system to ensure a clear and accurate understanding.