Is there an accepted definition of coprimality in commutative ring theory? I can think of at least three possible definitions of coprimality in commutative ring theory: call $a,b \in R$ are coprime iff


*

*if $c \mid a$ and $c \mid b$, then $c \mid 1$.

*if $a \mid c$ and $b \mid c$, then $ab \mid c$.

*$1$ can be written as a linear combination of $a$ and $b$, and hence every $r \in R$ can be written as a linear combination of $a$ and $b$.


These all agree for $\mathbb{Z}$.
I remark that (1) makes sense in an arbitrary poset, (2) makes sense in an arbitrary commutative monoid, and (3) makes sense in an arbitrary commutative ring.

Question. Is there an accepted definition of coprimality in commutative ring theory? If not, is there at least an accepted definition in principal ideal commutative rings?

Addendum 0. Here's another possible definition: call $a$ and $b$ coprime iff for all $a_0,a_1 \in R$ that divide $a$, and all $b_0,b_1 \in R$ that divide $b$, we have: $$a_0b_0 \sim a_1b_1 \rightarrow a_0 \sim a_1 \wedge b_0 \sim b_1$$
Addendum 1. Here's another one: call $a$ and $b$ coprime iff for all $a',b' \in R$, and all $r \in R$, we have that if $r(a',b') = (a,b),$ then $r$ is a unit.
Addendum 2. Darnit, here's another one: call $a$ and $b$ coprime iff $a$ is a unit in $R/bR$.
 A: Following your comment, here's an attempt to explain why the third definition is geometrically reasonable. (Note that I also don't really know anything about algebraic geometry.)
A fundamental idea was to think of a commutative ring as a geometric object, so that its element are really functions on some space. Formally, one associates with a commutative ring $R$ the pair $(\operatorname{Spec}R,R)$ where the former is the prime spectrum of $R$, consisting of its prime ideals with the Zariski topology. Modulo sheaves, this is the definition of the affine scheme associated to $R$. The prime spectrum is a contravariant functor from commutative rings to topological spaces defined by (set theoretic) pullback of prime ideals. An arrow of affine schemes is a pair of arrows going in opposite directions in the categories of spaces and commutative rings. This defines the category of affine schemes, and it's anti-equivalent to the category of commutative rings. Though seemingly formal, these definitions capture a lot of geometry, which you can read about in introduction to algebraic geometry.
Now ideals come into the picture: there's an inclusion-reversing isomorphism between the lattice of ideas of a commutative ring $R$ and the set of closed affine subschemes (defined as you can guess) of $(\operatorname{Spec}R,R)$. The bijection part follows purely formally, I think, from the fact equivalence relations are effective in the category of commutative rings. The inclusion reversal follows from some calculations involving properties of prime ideals.
Thus, given closed affine subschemes $Y,Z$ of an affine scheme $X$, it's reasonable to define $$Y\cup Z=(\operatorname{Spec} R/(I\wedge J),R/(I\wedge J))$$ and $$Y\cap Z=(\operatorname{Spec} R/(I\vee J),R/(I\vee J)).$$
Coprimality should mean two things are "separate from each other", an in light of this well motivated definition, the only real option is the third definition (in light of this comment).
Geometrically, the first definition seems artificial since there's no reason to even say the word 'principal' in such generality.
The equivalence of the second definition is partially characterized in (4) of this MO answer, but in general, it seem not to have too much to do with things being "separate". For geometric intuition for the product of ideals, see e.g this MO answer.
