Editing Commutative ring (section)

=== Ideals ===
{{Main|Ideal (ring theory)|l1=Ideal|Factor ring}}
''Ideals'' of a ring <math> R </math> are the [[submodule]]s of <math> R </math>, i.e., the modules contained in <math> R </math>. In more detail, an ideal <math> I </math> is a non-empty subset of <math> R </math> such that for all <math> r </math> in <math> R </math>, <math> i </math> and <math> j </math> in <math> I </math>, both <math> ri </math> and <math> i+j </math> are in <math> I </math>. For various applications, understanding the ideals of a ring is of particular importance, but often one proceeds by studying modules in general.

Any ring has two ideals, namely the [[0 (number)|zero ideal]] <math> \left\{0\right\} </math> and <math> R </math>, the whole ring. These two ideals are the only ones precisely if <math> R </math> is a field. Given any subset <math> F=\left\{f_j\right\}_{j \in J} </math> of <math> R </math> (where <math> J </math> is some index set), the ideal ''generated by'' <math> F </math> is the smallest ideal that contains <math> F </math>. Equivalently, it is given by finite [[linear combination]]s
<math display="block"> r_1 f_1 + r_2 f_2 + \dots + r_n f_n .</math>

==== Principal ideal domains ====
If <math> F </math> consists of a single element <math> r </math>, the ideal generated by <math> F </math> consists of the multiples of <math> r </math>, i.e., the elements of the form <math> rs </math> for arbitrary elements <math> s </math>. Such an ideal is called a [[principal ideal]]. If every ideal is a principal ideal, <math> R </math> is called a [[principal ideal ring]]; two important cases are <math> \mathbb{Z} </math> and <math> k \left[X\right] </math>, the polynomial ring over a field <math> k </math>. These two are in addition domains, so they are called [[principal ideal domain]]s.

Unlike for general rings, for a principal ideal domain, the properties of individual elements are strongly tied to the properties of the ring as a whole. For example, any principal ideal domain <math> R </math> is a [[unique factorization domain]] (UFD) which means that any element is a product of irreducible elements, in a (up to reordering of factors) unique way. Here, an element <math> a </math> in a domain is called [[irreducible element|irreducible]] if the only way of expressing it as a product
<math display="block"> a=bc ,</math>
is by either <math> b </math> or <math> c </math> being a unit. An example, important in [[Field (mathematics)|field theory]], are [[irreducible polynomial]]s, i.e., irreducible elements in <math> k \left[X\right] </math>, for a field <math> k </math>. The fact that <math> \mathbb{Z} </math> is a UFD can be stated more elementarily by saying that any natural number can be uniquely decomposed as product of powers of prime numbers. It is also known as the [[fundamental theorem of arithmetic]].

An element <math> a </math> is a [[prime element]] if whenever <math> a </math> divides a product <math> bc </math>, <math> a </math> divides <math> b </math> or <math> c </math>. In a domain, being prime implies being irreducible. The converse is true in a unique factorization domain, but false in general.

==== Factor ring ====
The definition of ideals is such that "dividing" <math> I </math> "out" gives another ring, the ''factor ring'' <math> R / I </math>: it is the set of [[coset]]s of <math> I </math> together with the operations
<math display="block"> \left(a+I\right)+\left(b+I\right)=\left(a+b\right)+I </math> and <math> \left(a+I\right) \left(b+I\right)=ab+I </math>.
For example, the ring <math> \mathbb{Z}/n\mathbb{Z} </math> (also denoted <math> \mathbb{Z}_n </math>), where <math> n </math> is an integer, is the ring of integers modulo <math> n </math>. It is the basis of [[modular arithmetic]].

An ideal is ''proper'' if it is strictly smaller than the whole ring. An ideal that is not strictly contained in any proper ideal is called [[maximal ideal|maximal]]. An ideal <math> m </math> is maximal [[if and only if]] <math> R / m </math> is a field. Except for the [[zero ring]], any ring (with identity) possesses at least one maximal ideal; this follows from [[Zorn's lemma]].