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Equation Commutative Ring Set Number Theory Ring Equivalence Of Forcing Notions Number Field Dedekind Domain U F D

1	Set
1	Equivalence Of Forcing Notions
1	Commutative Ring
1	Ring
1	Number Theory
1	Equation
2	U F D
2	Number Field
3	Dedekind Domain

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Ideal

Let be a ring. A left ideal (resp., right ideal) of is a nonempty subset $I \subset R$ such that:

$a-b \in I$ for all $a,b \in I$
$r \cdot a \in I$ (resp. $a \cdot r \in I$ ) for all $a \in I$ and $r \in R$

A two-sided ideal is a left ideal

which is also a right ideal. If

is a commutative ring, then these three notions of ideal are equivalent. Usually, the word “ideal” by itself means two-sided ideal.

The name “ideal” comes from the study of number theory. When the failure of unique factorization in number fields was first noticed, one of the solutions was to work with so-called “ideal numbers” in which unique factorization did hold. These “ideal numbers” were in fact ideals, and in Dedekind domains, unique factorization of ideals does indeed hold. The term “ideal number” is no longer used; the term “ideal” has replaced and generalized it.