Example

(1) Every subgroup of the additive group $\mathbf{Z}$ which contains 1 is equal to $\mathbf{Z}$. Thus $\mathbf{Z}$ is the only subring of $\mathbf{Z}$.

(2) Let $A$ be a ring and $(A_t)_{t \in I}$ a family of subrings of $A$; it is immediate that $\bigcap_{t \in I} A_t$ is a subring of $A$. In particular, the intersection of the subrings of $A$ containing a subset $X$ of $A$ is a subring called the subring of $A$ generated by $X$.

(3) Let $X$ be a subset of a ring $A$. The centralizer of $X$ in $A$ is a subring of $A$. In particular, the centre of $A$ is a subring of $A$.

(4) Let $G$ be a commutative group with operators; let $\Omega$ denote the set of operators and $\alpha \mapsto f_\alpha$ the action of $\Omega$ on $G$. Let $E$ be the endomorphism ring of the group without operators $G$ and $F$ the set of endomorphisms of the group with operators $G$. By definition, $F$ consists of the endomorphisms $\phi$ of $G$ such that $\phi \cdot f_\alpha = f_\alpha \cdot \phi$ for all $\alpha \in \Omega$. Therefore $F$ is a subring of the ring $E$. $F$ is called the endomorphism ring of the group with operators $G$ (cf. II, § 1, no. 2). Let $F_1$ be the subring of $E$ generated by the $f_\alpha$. Then $F$ is the centralizer of $F_1$ in $E$.