Order (ring theory)

Remarks

Example: $\Lambda = \Z[i]$ in $A = \Q(i)$, with $R = \Z$. It is a subring containing $1$, it is generated as a $\Z$-module by $1$ and $i$, and $1, i$ is a $\Q$-basis of $\Q(i)$. Other examples are $M_n(\Z)$ in $M_n(\Q)$ and the group ring $\Z G$ of a finite group $G$ in $\Q G$. Non-example: $\Z[\tfrac{1}{2}]$ is a subring of $\Q$ that spans $\Q$, but it is not a finitely generated $\Z$-module, so clause 1 fails. Non-example: $\Z$ inside $\Q(i)$ is finitely generated but fails clause 2, since $\Q \cdot \Z = \Q \ne \Q(i)$. When $R$ is a principal ideal domain, clause 1 makes $\Lambda$ a free $R$-module, and clause 2 then says its rank is $\dim_K A$. Every element of an order is integral over $R$.