Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept generalizes in an appropriate way some important properties of integers like "even number" or "multiple of 3".

Related Topics:
Ring theory - Abstract algebra - Subset - Ring - Integer

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For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of coprime numbers, and one can prove a generalized Chinese remainder theorem about ideals. In a certain class of rings important in number theory, the Dedekind domains, one can even recover a version of the fundamental theorem of arithmetic: in these rings, every nonzero ideal can be uniquely written as a product of prime ideals.

Related Topics:
Prime ideal - Prime number - Coprime - Chinese remainder theorem - Number theory - Dedekind domain - Fundamental theorem of arithmetic

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An ideal can be used to construct a factor ring in a similar way as a normal subgroup in group theory can be used to construct a factor group. The concept of an order ideal in order theory is derived from the notion of ideal in ring theory.

Related Topics:
Normal subgroup - Group theory - Factor group - Order ideal - Order theory

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