Tips on proving that a set has an algebraic structure

If we read the article about ring on wikipedia, we can see that the properties of ring are as follows.

A ring is a set $R$ equipped with two binary operations $+$ (addition) and $⋅$ (multiplication) satisfying the following three sets of axioms, called the ring axioms:

1. $R$ is an abelian group under addition, meaning that:

• $(a + b) + c = a + (b + c)$ for all $a, b, c$ in $R$ (that is, $+$ is associative).
• $a + b = b + a$ for all $a, b$ in $R$ (that is, $+$ is commutative).
• There is an element $0$ in $R$ such that $a + 0 = a$ for all $a$ in $R$ (that is, $0$ is the additive identity).
• For each $a$ in $R$ there exists $−a$ in $R$ such that $a + (−a) = 0$ (that is, $−a$ is the additive inverse of $a$).

2. $R$ is a monoid under multiplication, meaning that:

• $(a · b) · c = a · (b · c)$ for all $a, b, c$ in $R$ (that is, $⋅$ is associative).
• There is an element $1$ in $R$ such that $a · 1 = a$ and $1 · a = a$ for all a in $R$ (that is, $1$ is the multiplicative identity).

3. Multiplication is distributive with respect to addition, meaning that:

• $a · (b + c) = (a · b) + (a · c)$ for all $a, b, c$ in $R$ (left distributivity).
• $(b + c) · a = (b · a) + (c · a)$ for all $a, b, c$ in $R$ (right distributivity).

And to check whether a set is a ring, we have to check one by one whether it satisfies the properties written above. And a typical example of a ring is the integer $\mathbb{Z}$. So I thought about an algorithm to check if something is a ring and practiced with $\mathbb{Z}$ to check the properties of rings. One of those results is shown below as an example.

If $a,b,c \in \mathbb{Z}$, then $(a + b) + c = a + (b + c)$.

I tried to prove that the set of integers $\mathbb {Z}$ is a ring for practice, but as above, I only got very vague results. Is there a way to avoid such vague results and get rigorous results in the process of proving that a set is a ring? I think that proving that a set has an algebraic structure such as a group, ring, or field in this way is too vague, so I would like to get help.

Edit: Of course, We can intuitively know that the integer system satisfies all the properties of the ring written above, but I think that proof is a task of thoroughly investigating the obvious, so I feel a little uncomfortable about just passing it on as if it satisfies these properties just because it is an integer. Is this the best proof that the set of integers is a ring?

And when proving that a set other than a number system has an algebraic structure such as a group, ring, or field, I have to prove that it satisfies the properties that each algebraic structure must have, but I am still immature, so it is difficult to prove. Can you give me some tips for proofs on general sets?