Pinter's "A Book of Abstract Algebra" defines a ring $A$ stipulating the following $3$ axioms:
- $A$ with addition alone is an abelian group.
- Multiplication is associative.
- Multiplication is distributive over addition.
I am a little confused as to why the third axiom is needed. Doesn't axiom $1$ naturally lead to $3$? Consider the following example:
$(a \circ_+ b)^5 = (a\circ_+ b) \circ_+ (a\circ_+ b) \circ_+ (a\circ_+ b) \circ_+ (a\circ_+ b) \circ_+ (a\circ_+ b)$
Because the operation of addition is an abelian group, the above statement is equal to:
$a^5 \circ_+ b^5$
if we simply move the notation around...doesn't this read just like:
$5(a\circ_+ b)=5(a) \circ_+ 5(b)$ ...which could be generalized to any number.
From this explanation, it seems that multiplication's distributive property over addition is already baked into the system without the need to specify it.
What am I missing / incorrectly concluding? Cheers~