Editing Ring (mathematics) (section)

=== Ideal ===
{{main|Ideal (ring theory)}}
Let {{mvar|R}} be a ring.  A '''left ideal''' of {{mvar|R}} is a nonempty subset {{mvar|I}} of {{mvar|R}} such that for any {{mvar|x, y}} in {{mvar|I}} and {{mvar|r}} in {{mvar|R}}, the elements {{math|''x'' + ''y''}} and {{mvar|rx}} are in {{mvar|I}}. If {{mvar|R I}} denotes the {{mvar|R}}-span of {{mvar|I}}, that is, the set of finite sums
: <math>r_1 x_1 + \cdots + r_n x_n \quad \textrm{such}\;\textrm{that}\; r_i \in R  \; \textrm{ and } \; x_i \in I,</math>
then {{mvar|I}} is a left ideal if {{math|''RI'' ⊆ ''I''}}. Similarly, a '''right ideal''' is a subset {{mvar|I}} such that {{math|''IR'' ⊆ ''I''}}. A subset {{mvar|I}} is said to be a '''two-sided ideal''' or simply '''ideal''' if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of {{mvar|R}}. If {{mvar|E}} is a subset of {{mvar|R}}, then {{math|''RE''}} is a left ideal, called the left ideal generated by {{mvar|E}}; it is the smallest left ideal containing {{mvar|E}}. Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of {{mvar|R}}.

If {{mvar|x}} is in {{mvar|R}}, then {{math|''Rx''}} and {{math|''xR''}} are left ideals and right ideals, respectively; they are called the [[principal ideal|principal]] left ideals and right ideals generated by {{mvar|x}}. The principal ideal {{math|''RxR''}} is written as {{math|(''x'')}}. For example, the set of all positive and negative multiples of {{math|2}} along with {{math|0}} form an ideal of the integers, and this ideal is generated by the integer&nbsp;{{math|2}}. In fact, every ideal of the ring of integers is principal.

Like a group, a ring is said to be [[simple ring|simple]] if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field.

Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite [[total order#Chains|chain]] of left ideals is called a left [[Noetherian ring]]. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left [[Artinian ring]]. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the [[Hopkins–Levitzki theorem]]). The integers, however, form a Noetherian ring which is not Artinian.

For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal {{mvar|P}} of {{mvar|R}} is called a [[prime ideal]] if for any elements <math>x, y\in R</math> we have that <math>xy \in P</math> implies either <math>x \in P</math> or <math>y\in P.</math> Equivalently, {{mvar|P}} is prime if for any ideals {{math|''I''}}, {{math|''J''}} we have that {{math|''IJ'' ⊆ ''P''}} implies either {{math|''I'' ⊆ ''P''}} or {{math|''J'' ⊆ ''P''}}. This latter formulation illustrates the idea of ideals as generalizations of elements.