Commutative ring

A commutative ring is a ring $R$ such that $ab=ba$ for all $a,b\in R$.

Commutative rings support a particularly rich theory of ideals and spectrum-type constructions; many foundational results are phrased in terms of the commutative ring axiom . In commutative rings, left/right distinctions for ideals and zero divisors disappear.

Examples:

• $\mathbb Z$ and $\mathbb Q$ are commutative rings.
• For a field $k$, the polynomial ring $k[x_1,\dots,x_n]$ is commutative.
• $M_n(k)$ for $n\ge 2$ is not commutative (matrix multiplication generally fails to commute).