Divisibility

Note that this definition exists in the absence of division, and no division operation need be defined for divisibility to be well defined. This leads to the fact that $0\mid 0$ even though $0÷0$ is not defined.

There are many useful properties of divisibility, some of which apply generally, while others require additional restrictions on the underlying ring.

Proof

These facts follow very simply from the explicit form of ideals generated by subsets. That is:

since $an$ for $n\in R$ is the form of all elements in $⟨a⟩$ when working in a commutative ring with identity. The requirement of a ring with identity is used in applying the result linked above.

Then, we have that:

as a direct consequence of this result.

Note specifically the use of a commutative ring, which means that we also have $na=b$. Divisibility is rarely defined in non-commutative rings, although one can construct an analogous definition by distinguishing between left and right divisibility as follows.