I'm going through this proof in Allan Clark's Elements of Abstract Algebra to prove that a primigenial ring is a Dedekind Domain.(adsbygoogle = window.adsbygoogle || []).push({});

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!

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# Primigenial ring ideals question

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