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!

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Primigenial ring ideals question

**Physics Forums | Science Articles, Homework Help, Discussion**