ok, suppose in a ring R a|b. what does this mean? it means there is some c in R such that b = ca (we are going to assume commutativity, the non-commutative case gets too involved).

well if b = ca, this means b is in the ideal generated by a, (a), and hence (b) ⊆ (a).

on the other hand suppose that for a,b in R, (b) ⊆ (a). this means that b = ra, for some r in R, so that a|b.

that is: "b divisible by a" is equivalent to (b) ⊆ (a).

now suppose p in R is a prime element. this means that if p|(ab), then either p|a, or p|b. equivalently if ab is in (p), then either a is in (p), or b is in (p).

this means a prime element generates a prime ideal.

the proviso that p not be a unit, is to ensure that (p) ≠ R, because trivially ab in R implies a in R AND b in R (in other words, only PROPER ideals can be prime, or else they wouldn't tell us anything).

for commutative rings R, (a,b,...,k) can be thought of as R-linear combinations of {a,b,...,k}. if we have just a 1-element set, like {a}, then (a) is "R-multiples of a".

saying a|b is the same as saying: b is an "R-multiple" of a. that's why we study IDEALS, for any element x in an ideal I, it also contains all "R-multiples" of x.

this situation is particularly handy with principal ideal domains, because then there is a 1-1 correspondence between "prime elements" and "prime ideals".

now: why integral domains?

well, suppose R has a zero divisor a. then there is some b ≠ 0, with ab = 0. now: consider (ab) = (0) = {0}. clearly neither a nor b is in (0) = {0}, so {0} is not a prime ideal. why is this bad?

because 0 is an element of EVERY IDEAL so for an ideal to be prime in a ring with zero divisors it needs to contain a lot of zero divisors.

for example, since 2*3 = 0 in Z_{6}, any prime ideal of Z_{6}has to either contain ALL multiples of 2 or ALL multiples of 3, or else we wouldn't have one at all.

in general, in rings, we don't have a linear order, so we tend to order elements of rings by divisibility. zero divisors sort of mess with this (but it IS possible to have prime ideals in rings with zero divisors).