I'm going through this proof in Allan Clark's Elements of Abstract Algebra to prove that a primigenial ring is a Dedekind Domain. A primigenial ring is one in which every proper ideal can be written as a product of proper prime ideals. There's a step in the proof that I'm not able to understand... We're given p an invertible proper prime ideal in a primigenial ring R. a is an element in R - p. He proves that p + (a) = p^2 + (a) then... [tex]p = p \cap (p^2 + (a))[/tex]... no problem here... then the next step: [tex]p \cap (p^2 + (a)) \subset p^2 + (a)p[/tex]. I'm not sure how he does this step... I'd appreciate any help or hints. Thanks a bunch!