Ideals are special subsets of rings, which are closed under the addition operation an absorb other elements of the ring, when multiplied by elements of the ideal. As an example, the set \(E\) of even numbers in the ring \((\mathbb Z, + ,\cdot)\) constitutes an ideal, since the addition and subtraction of any two even numbers preserves evenness, and multiplying any (even or odd) integer by an even number also creates an even number.
Table of Contents
- Definition: Ideal
- Proposition: Principal Ideals being Prime Ideals
- Proposition: Principal Ideals being Maximal Ideals
- Definition: Principal Ideal
- Definition: Generating Set of an Ideal
- Definition: Prime Ideal
- Definition: Maximal Ideal
- Lemma: Prime Ideals of Multiplicative Systems in Integral Domains
- Theorem: Connection between Rings, Ideals, and Fields
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