More precisely, why is it that all rings are required by the axioms to have commutativity in addition, but are not held to the same axiom regarding multiplication? I know that we have commutative and non-commutative rings depending on whether or not they are commutative in multiplication, but I am wondering why it is that the axioms were defined that way, providing us with this option.
I am using this list of axioms, from David Sharpe’s Rings and factorization:
Definition 1.3.1. A ring is a non-empty set $R$ which satisfies the following axioms:
(1) $R$ has a binary operation denoted by $+$ defined on it;
(2) addition is associative, i.e. \begin{align} a + \left(b+c\right) = \left(a+b\right) + c \text{ for all } a, b, c \in R \end{align} (so that we can write $a+b+c$ without brackets);
(3) addition is commutative, i.e. \begin{align} a + b = b + a \text{ for all } a, b \in R; \end{align}
(4) there is an element denoted by $0$ in $R$ such that \begin{align} 0 + a = a \text{ for all } a \in R \end{align} (there is only one such element because, if $0_1$ and $0_2$ are two such, then $0_1 = 0_1 + 0_2 = 0_2$ and they are the same -- we call $0$ the zero element of $R$);
(5) for every $a \in R$, there exists an element $-a \in R$ such that \begin{align} \left(-a\right) + a = 0 \end{align} (there is only one such element for each $a$, because if $b + a = 0$ and $c + a = 0$, then \begin{align} b = 0 + b = \left(c + a\right) + b = c + \left(a + b\right) = c + 0 = c; \end{align} we call $-a$ the negative of $a$);
(6) $R$ has a binary operation denoted by multiplication defined on it;
(7) multiplication is associative, i.e. \begin{align} a\left(bc\right) = \left(ab\right)c \text{ for all } a, b, c \in R; \end{align}
(8) multiplication is left and right distributive over addition, i.e. \begin{align} a\left(b+c\right) = ab + ac,\ \left(a+b\right)c = ac + bc \text{ for all } a, b, c \in R; \end{align}
(9) there is an element denoted by $1$ in $R$ such that $1 \neq 0$ and \begin{align} 1 \cdot a = a \cdot 1 = a \text{ for all } a \in R \end{align} (as for the zero element, there is only one such element, and it is called the identity element of $R$).