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Knowing that the monic polynomial and finitely-generated module definitions of integral elements over a commutative ring are equivalent, how can one prove that these elements form a ring?

Thanks.

eduardogbg
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  • If you read the solution at the linked question, it actually does it for integral elements too. – rschwieb Feb 01 '18 at 22:35
  • Which linked question? – Bernard Feb 01 '18 at 22:43
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    This is not exactly the same question: the O.P. doesn't ask for explicit polynomials with roots the sum and product of two integral elements, but how to show these sum and product are integral elements. – Bernard Feb 01 '18 at 22:47
  • @Bernard I don't contend that it's the same question, but I still think (as the template says) the question is answered there, and/or/in combination with at the two links in the problem statement. It could also perhaps be a case of me picking a bad duplicate. I can't believe this question hasn't been asked here before... – rschwieb Feb 02 '18 at 15:29

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