An object $I$ in a category $\mathcal{A}$ is initial if for each $\mathcal{A}$-object $X$, there is a unique morphism $I\rightarrow X$.
An object $J$ in a category $\mathcal{A}$ is final if for each $\mathcal{A}$-object $X$, there is a unique morphism $X\rightarrow J$.
These are definitions from Cohn's Basic Algebra.
Let $\mathcal{R}$ be the category of rings. In this category $0$ ring is final object, and $\mathbb{Z}$ is initial object.
Q. In defining rings, the author mentioned that it should contain $1$, multiplicative identity; it is not necessarily distinguished from $0$. The author call $0$ to be trivial ring (see p. 79, Section 4.1). In defining ring homomorphism, author says that the multiplicative identity should go to multiplicative identity. I am not getting then why the zero ring can not be initial object? The map $0\mapsto 0\in R$ is unique ring homomorphism from ring $\{0\}$ to any ring $R$, am I right?
Best Answer
The function $f:\{0\}\to R$ with $f(0) = 0$ doesn't map the multiplicative identity of $\{0\}$ to the multiplicative identity of $R$ (unless $R$ is also a zero ring).
$\Bbb Z$ is the initial object of rings with multiplicative identity, as $f:\Bbb Z\to R$ with $f(n) = n\cdot 1_R$ is the unique unit-preserving ring homomorphism from $\Bbb Z$ to $R$.