Ring

A ring is a set R with two binary operations defined on it: addition , denoted by +, and multiplication, denoted by ⋅, or just by juxtaposition, satisfying the following six axioms:

1. Addition is commutative: a + b = b + a for each pair a, b in R.
2. Addition is associative: a + (b + c) = (a + b) + c for a, b, c ∈ R.
3. There exists an additive identity, denoted by 0, such that a + 0 = a for each a ∈ R.
4. For each a ∈ R, there exists an additive inverse, denoted by −a, such that a + (−a) = 0.
5. Multiplication is associative: a(bc) = (ab)c for a, b, c ∈ R.
6. Multiplication is left and right distributive over addition: a(b + c) = ab + ac, and (b + c)a = ba + ca for a, b, c ∈ R.

The ring R is commutative if

1. Multiplication is commutative: ab = ba for a, b in R.

We call R a ring with identity if

1. There exists a multiplicative identity denoted by 1 such that a ⋅ 1 = a and 1 ⋅ a = a for each a in R.

If R satisfies (1) through (8), then R is a commutative ring with identity.

(Rosenberger, Schürenberg, and Wienke 2024, 2 chap.1)