Commutative Rings

Recall that if $(G, *)$ is a group with the additional property that for all $x, y \in G$ we have that $x * y = y * x$ then $(G, *)$ is said to be an abelian group or a commutative group.

Now suppose that we have a ring, $(R, +, *)$ Then $(R, +)$ by definition is an abelian group, but there's no requirement that the operation $*$ needs to be commutative as well Rings for which both operations are commutative are called commutative rings which we formally define below.

Definition: Let $(R, +, *)$ be a ring. Then $(R, +, *)$ is said to be a Commutative Ring if for all $x, y \in R$ we have that $x * y = y * x$, i.e., the operation $*$ is commutative.

There are many examples of commutative rings. For example, $(\mathbb{R}, +, *)$ is a commutative ring, and of course, $(\mathbb{Z}, +, *)$, and $(\mathbb{Q}, +, *)$. This is because multiplication of the real numbers (and the integers/rationals) is commutative.

Of course, not all rings are commutative though. For example, the ring $(M_{nn}, +, *)$ where $M_{nn}$ denotes the set of all $n \times n$ matrices is not commutative for $n \geq 2$ because matrix multiplicative is not commutative in general.