Let $R$ be a commutative ring and $E$ an extension ring. If $\alpha \in E$ , and commutes with all elements of $R$ then the smallest subring of $E$ containing $R$ and $\alpha$ is denoted by $R[\alpha]$ We say that $R[\alpha]$ is obtained from $R$ by adjoining $\alpha$ to $R$ via ring adjunction.

By the Theorem 1 about ``evaluation homomorphism'', $$R[\alpha] = \{f(\alpha)\mid \, f(X)\in R[X]\},$$ where $R[X]$ is the polynomial ring in one indeterminate over $R$ Therefore, $R[\alpha]$ consists of all expressions which can be formed of $\alpha$ and elements of the ring $R$ by using additions, subtractions and multiplications.

Examples: The polynomial rings $R[X]$ the ring $\mathbb{Z}[i]$ of the Gaussian integers, the ring $\mathbb{Z}[\frac{-1+i\sqrt{3}}{2}]$ of Eisenstein integers.