Definition: Ideal

Let \((R, + , \cdot)\) be a ring. An left (right) ideal is a subset of \(R\) a with the following properties:

  1. \((I, +)\) is a subgroup of \((R, +)\); equivalently, \(I\) is not empty and for \(a,b\in I\) we have \(a - b\in I\).
  2. \(\forall i \in I, \forall r \in R: \quad r \cdot i \in I\) (for a right ideal)
  3. \(\forall i \in I, \forall r \in R: \quad i \cdot r \in I\) (for a left ideal)

If \(I\) is both, a left and a right ideal, we call \(I\) simply an ideal and write \(I\lhd R\).

  1. Lemma: Greatest Common Divisor and Least Common Multiple of Ideals
  2. Definition: Divisibility of Ideals
  3. Definition: Addition of Ideals

Definitions: 1 2 3 4 5 6 7 8
Lemmas: 9 10 11
Proofs: 12 13 14 15
Propositions: 16 17
Theorems: 18

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  1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013