Let $R$ be a commutative ring and $E$ an extension ring of it.  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.
 $R[\alpha]=\{f(\alpha)\mid\,f(X)\in R[X]\},$
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.