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I've been doing some computations with ideals in polynomial rings given by a finite set of generators, and I'd like to know if there are some basic facts that can make my life easier. For example, I want to say that the following are true in a commutative ring:

  • $(a_1,\ldots,a_n) + (b_1,\ldots,b_m) = (a_1,\ldots,a_n,b_1,\ldots,b_m)$
  • $(a_1,\ldots,a_n)(b_1,\ldots,b_m) = (\{a_ib_j : 1 \leq i \leq n, 1 \leq j \leq m\})$.

Do those two identities always hold, and are there other ones I should know? (For example I can't figure out a way to write the intersection of two finitely generated ideals in terms of their generators, or similarly for the ideal quotient or radical.)

Nick A.
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    Yes, those are true. Other operations such as intersection are harder; for example in polynomial ring they can be done using Groebner bases: https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis – Angina Seng Sep 21 '20 at 01:36

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