If you want an easy to understand new concrete example of associative rules, distributive rule, and multiplication inverses and identity rules that contribute to defining a ring, consider the so-called "tropical semiring", i.e. min-addition semiring over the reals. For two real numbers, "addition" of $x,y$ is defined as $\min(x,y)$, and "multiplication" of two reals is defined as $x + y$. With this definition of addition and multiplication, you can check that associativity holds for addition and multiplication, and that the distributive property holds, and that $0$ is the multiplicative identity, and multiplicative inverses exist (the inverse of $x$ is $-x$). However there is no additive identity and no additive inverses, because min is an "irreversible" operation. Hence why it's called a "semi-ring" instead of a ring. But it illustrates most of the ring properties and is very easy to understand and verify.