Primary decomposition

Hi sir,

just a question regarding the proofs required. for the primary decomposition theorem, we have used the lemma:

if f and g are coprime then ker(fg(T))=Ker(f(T)) direct+ Ker(g(T)).

If an exam question asks us to prove the primary decomposition theorem do we need to state and prove the above lemma as well or can we just state it and assume that it is true?

Thanks for your help!

Reply:

It’s hard to give an entirely straightforward answer to a question of this sort. This is because the sufficiency of an argument depends strongly on the overall structure of what’s written, and the degree of understanding that appears globally. But I can try be a bit more specific in this instance. The statement you cite can be deduced easily from Bezout’s lemma. So if you would write down how this deduction goes, citing Bezout’s lemma in the appropriate place, then I would be inclined to give full credit for that portion. Once again, this is contingent on presenting the whole thing in a clear and concise way that displays definite understanding.

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