Unital ring

A unital ring is a ring $R$ together with an element $1\in R$ such that $1a=a1=a$ for all $a\in R$.

In a unital ring, elements that admit multiplicative inverses are units , and they form the group of units . Many constructions (e.g. polynomial rings) and many definitions of ring homomorphisms are most natural in the unital setting.

Examples:

• $\mathbb Z$ is unital with identity $1$.
• $M_n(R)$ is unital for any unital ring $R$, with identity matrix $I_n$.
• $2\mathbb Z$ is a ring but not unital (its inherited multiplication has no identity element).