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I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.1 Prime Ideals and Maximal Ideals ...
I need some help with understanding the proof of Theorem 5.12 ... ...Theorem 5.12 reads as follows:View attachment 5940
In the above text Rotman writes the following:" ... ... If $$(p) \subseteq J = (a)$$, then $$a|p$$. Hence either $$a$$ and $$p$$ are associates, in which case $$(a) = (p)$$, or $$a$$ is a unit, in which case $$J = (a) = R$$. ... ... ... "My question is as follows:Rotman argues, (as I interpret his argument), that $$a|p$$ implies that either $$a$$ and $$p$$ are associates ... or ... $$a$$ is a unit ...Can someone please explain (slowly and clearly :) ) why this is the case ... ... ?Hope someone can help ... ...
Peter
I need some help with understanding the proof of Theorem 5.12 ... ...Theorem 5.12 reads as follows:View attachment 5940
In the above text Rotman writes the following:" ... ... If $$(p) \subseteq J = (a)$$, then $$a|p$$. Hence either $$a$$ and $$p$$ are associates, in which case $$(a) = (p)$$, or $$a$$ is a unit, in which case $$J = (a) = R$$. ... ... ... "My question is as follows:Rotman argues, (as I interpret his argument), that $$a|p$$ implies that either $$a$$ and $$p$$ are associates ... or ... $$a$$ is a unit ...Can someone please explain (slowly and clearly :) ) why this is the case ... ... ?Hope someone can help ... ...
Peter