Let $R$ denote a commutative ring with unity. Consider elements $a,b \in R$. Is there an accepted notation (like $a \| b$ or some such) for the number of times that $a$ divides $b$? Explicitly, we can define $a \| b \in \mathbb{N} \cup \{\infty\}$ as follows.
- $a \| b$ is the maximum $k \in \mathbb{N}$ satisfying $a^k \mid b$, as long as the set of all $k \in \mathbb{N}$ such that $a^k \mid b$ is bounded.
- $a \| b=\infty$ otherwise.
I would also like to know the name of this "$\mathbb{N}$-valued relation", if it has one.
One motivation is that if $R$ is a UFD and we're given $a,b \in R$, then if $a$ is irreducible, it follows that $a \| b$ is the number of occurrences of $a$ in the factorization of $b$ into irreducible elements (and $a \| b$ is necessarily finite in this case).
Another motivation (observation, really) is that $\|$ behaves a bit like a preorder. In place of reflexivity, we have $1 \leq (a \| a),$ and in place of transitivity, we have $(a \| b)(b \| c) \leq (a \| c)$.
For $R$ a UFD and $a$ irreducible (and hence prime), I would say $v_a(b)$ (the valuation of $b$ at $a$). This is defined as $b = a^{v_a(b)} u$, where $u$ is a unit wrt to $a$. For example, $R = \mathbb{Z}$, $v_3(2)=0$, $v_3(9)=2$, $v_3(21) = 1$.