Def: A valuation on a ring $A$ (commutative with identity) is a map $\lvert \cdot \rvert : A \to \Gamma \cup \{0\}$ such that for all $a,b\in A$,
$\lvert a+b \rvert \leq \max\{\lvert a \rvert, \lvert b \rvert \}$,
$\lvert ab \rvert=\lvert a \rvert\lvert b \rvert$,
$\lvert 0 \rvert=0$ and $\lvert 1 \rvert=1$,
where $\Gamma$ is a totally ordered group.
Def: Two valuations $\lvert \cdot \rvert_1,\lvert \cdot \rvert_2$ on $A$ are said to be equivalent if for every $a,b \in A$ we have that $\lvert a \rvert_1 \leq \lvert b\rvert_1$ if and only if $\lvert a \rvert_2 \leq \lvert b\rvert_2$.
I study valuations right now and saw the second definition. I tried to find examples about equivalent valuations but I couldn't. Can you help me please?
For two examples of equivalent valuations that are not identically equivalent, fix a prime $p \in \mathbb{Z}$ with $p \geq 2$ and consider the integers $\mathbb{Z}$ as our base ring. We will construct now the standard $p$-adic norm on $\mathbb{Z}$, $\lvert \cdot \rvert_p:\mathbb{Z} \to \mathbb{Q}$ defined by first considering the function $$ v_p(n) := \begin{cases} 0 & {\rm if}\, n = 0; \\ \max\lbrace m \in \mathbb{N} \; : \; p^m|n \rbrace & {\rm if}\, n \ne 0; \end{cases} $$ and then defining $\lvert \cdot \rvert_p$ by saying that $$ \lvert n \rvert_p = \begin{cases} 0 & {\rm if}\, n = 0; \\ p^{-v_p(n)} & {\rm if}\, n \ne 0. \end{cases} $$ for all $n \in \mathbb{Z}$. It can then be easily verified that $\lvert \cdot \rvert_p$ satisfies the properties you ask of it. Now, to get an equivalent valuation, fix a constant $a > 1$ for $a \in \mathbb{Z}$ and with $\gcd(a,p) =1$. We can define a map $\lVert \cdot\rVert_p:\mathbb{Z} \to \mathbb{Q}$ by defining $$ \lVert n \rVert_p := \begin{cases} 0 & {\rm if}\, n = 0; \\ a^{-v_p(n)} & {\rm if}\, n \ne 0. \end{cases} $$ It is then just as straightforward to show that $\lVert \cdot \rVert_p$ satisfies the properties you ask as well, but that $\lvert \cdot \rvert_p \ne \lVert \cdot \rVert_p$ by considering what each map does on $-p$. However, the maps are equivalent in the sense you request because since $a, p > 1$, a standard calculus argument shows that $$ \lvert n \rvert_p \leq \lvert m \rvert_p $$ if and only if $$ \lVert n \rVert_p \leq \lVert m \rVert_p. $$