Category of rings

Remarks

The ring $\Z$ is an initial object: for each ring $R$ the only homomorphism $\Z \to R$ sends $n$ to $n \cdot 1_R$. The zero ring is a terminal object, since every ring has exactly one map to it. The zero ring is not initial, because a homomorphism from it must send $1 = 0$ to $1_R = 0_R$, which fails when $R$ is nonzero; so $\mathbf{Ring}$ has no zero object. The product is the direct product $R \times S$ with componentwise operations. The commutative rings form a full subcategory $\mathbf{CRing}$, in which the coproduct of $R$ and $S$ is the tensor product $R \otimes_{\Z} S$. The inclusion $\Z \to \Q$ is an epimorphism of $\mathbf{Ring}$ that is not surjective: two homomorphisms out of $\Q$ that agree on $\Z$ agree on all fractions $m/n$.