"Mostly" true, but if you accept this statement as-is, it's a little misleading.
The integers (with standard addition/multiplication operators) are indeed a ring, but a ring is a very loose definition that does not reveal say too much about the integers themselves. There are several stronger algebraic structures than integers that don't have division.
The integers themselves are an example of a general algebraic structure called a "domain," (or, more commonly, "integral domain") which is a commutative ring (that is, commutative under multiplication as well), has distinct additive and multiplicative identities, and has the property that if a and b are integers and a*b = 0, then either a or b must be zero.
When mathematicians in the past were studying divisibility in the integers, they have generally studied domains, because they essentially isolate the division property among different operations. Even an integral domain is not the strongest structure you can place on integers. There are even more specific subclasses of integral domains that the integers fall under, but that's best left for a class on algebraic structures and not a comment on HN.
