Prove A:B is an ideal of R?



Answered question


Is A« B an ideal of R?


Answer & Explanation

Jeffrey Jordon

Jeffrey Jordon

Expert2022-11-08Added 2605 answers

You don’t need to show that ar,raA for all aA; nor that abA (for a,bA, I assume?). Since A is an ideal, this is given.

In order to prove that A:B is an ideal, you need to show that

  • 0A:B (or alternatively A:B, which you have already done),
  • for all x,yA:B, you have xyA:B (or alternatively x+yA:B), and
  • for all rR and xA:B, you have rxA:B and xrA:B.

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