Editing Ring (mathematics) (section)

== Basic concepts ==

=== Products and powers ===
For each nonnegative integer {{mvar|n}}, given a sequence {{tmath|1= (a_1,\dots,a_n) }} of {{mvar|n}} elements of {{mvar|R}}, one can define the product {{tmath|1= \textstyle P_n = \prod_{i=1}^n a_i }} recursively: let {{math|1=''P''<sub>0</sub> = 1}} and let {{math|1=''P''<sub>''m''</sub> = ''P''<sub>''m''−1</sub>''a''<sub>''m''</sub>}} for {{math|1 ≤ ''m'' ≤ ''n''}}.

As a special case, one can define nonnegative integer powers of an element {{mvar|a}} of a ring: {{math|1=''a''{{sup|0}} = 1}} and {{math|1=''a''{{sup|''n''}} = ''a''{{sup|''n''−1}}''a''}} for {{math|''n'' ≥ 1}}.  Then {{math|1=''a''<sup>''m''+''n''</sup> = ''a''<sup>''m''</sup>''a''<sup>''n''</sup>}} for all {{math|''m'', ''n'' ≥ 0}}.

=== Elements in a ring ===
A left [[zero divisor]] of a ring {{mvar|R}} is an element {{mvar|a}} in the ring such that there exists a nonzero element {{mvar|b}} of {{mvar|R}} such that {{math|1=''ab'' = 0}}.{{efn|Some other authors such as Lang further require a zero divisor to be nonzero.}} A right zero divisor is defined similarly.

A [[nilpotent element]] is an element {{mvar|a}} such that {{math|1=''a{{sup|n}}'' = 0}} for some {{math|''n'' > 0}}. One example of a nilpotent element is a [[nilpotent matrix]]. A nilpotent element in a [[zero ring|nonzero ring]] is necessarily a zero divisor.

An [[idempotent element (ring theory)|idempotent]] <math>e</math> is an element such that {{math|1=''e''{{sup|2}} = ''e''}}. One example of an idempotent element is a [[projection (linear algebra)|projection]] in linear algebra.

A [[unit (ring theory)|unit]] is an element {{mvar|a}} having a [[multiplicative inverse]]; in this case the inverse is unique, and is denoted by {{math|''a''{{sup|–1}}}}. The set of units of a ring is a [[group (mathematics)|group]] under ring multiplication; this group is denoted by {{math|''R''{{sup|×}}}} or {{math|''R''*}} or {{math|''U''(''R'')}}. For example, if {{mvar|R}} is the ring of all square matrices of size {{mvar|n}} over a field, then {{math|''R''{{sup|×}}}} consists of the set of all invertible matrices of size {{mvar|n}}, and is called the [[general linear group]].

=== Subring ===
{{main|Subring}}
A subset {{mvar|S}} of {{mvar|R}} is called a [[subring]] if any one of the following equivalent conditions holds:
* the addition and multiplication of {{mvar|R}} [[restricted function|restrict]] to give operations {{math|''S'' × ''S'' → ''S''}} making {{mvar|S}} a ring with the same multiplicative identity as&nbsp;{{mvar|R}}.
* {{math|1 ∈ ''S''}}; and for all {{mvar|x, y}} in {{mvar|S}}, the elements {{mvar|xy}}, {{math|''x'' + ''y''}}, and {{mvar|−x}} are in&nbsp;{{mvar|S}}.
* {{mvar|S}} can be equipped with operations making it a ring such that the inclusion map {{math|''S'' → ''R''}} is a ring homomorphism.

For example, the ring {{tmath|\Z}} of integers is a subring of the [[field (mathematics)|field]] of real numbers and also a subring of the ring of [[polynomial]]s {{tmath|\Z[X]}} (in both cases, {{tmath|\Z}} contains&nbsp;1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers {{tmath|2\Z}} does not contain the identity element {{math|1}} and thus does not qualify as a subring of&nbsp;{{tmath|\Z;}} one could call {{tmath|2\Z}} a [[rng (algebra)|subrng]], however.

An intersection of subrings is a subring. Given a subset {{mvar|E}} of {{mvar|R}}, the smallest subring of {{mvar|R}} containing {{mvar|E}} is the intersection of all subrings of {{mvar|R}} containing&nbsp;{{mvar|E}}, and it is called ''the subring generated by&nbsp;{{math|E}}''.

For a ring {{mvar|R}}, the smallest subring of {{mvar|R}} is called the ''characteristic subring'' of {{mvar|R}}. It can be generated through addition of copies of {{math|1}} and&nbsp;{{math|−1}}. It is possible that {{math|1=''n'' · 1 = 1 + 1 + ... + 1}} ({{mvar|n}} times) can be zero. If {{mvar|n}} is the smallest positive integer such that this occurs, then {{mvar|n}} is called the ''[[Characteristic (algebra)|characteristic]]'' of&nbsp;{{mvar|R}}. In some rings, {{math|''n'' · 1}} is never zero for any positive integer {{mvar|n}}, and those rings are said to have ''characteristic zero''.<!-- By using a homomorphism from the integers into {{mvar|R}} that sends {{math|1}} to {{math|1}}, it can be shown by an isomorphism theorem that the characteristic subring is always a quotient ring of the integers. However, we haven't introduced homomorphisms and quotient rings yet. -->

Given a ring {{mvar|R}}, let {{math|Z(''R'')}} denote the set of all elements {{mvar|x}} in {{mvar|R}} such that {{mvar|x}} commutes with every element in {{mvar|R}}: {{math|1=''xy'' = ''yx''}} for any {{mvar|y}} in&nbsp;{{mvar|R}}. Then {{math|Z(''R'')}} is a subring of&nbsp;{{mvar|R}}, called the [[Center (ring theory)|center]] of&nbsp;{{mvar|R}}. More generally, given a subset {{mvar|X}} of&nbsp;{{mvar|R}}, let {{mvar|S}} be the set of all elements in {{mvar|R}} that commute with every element in&nbsp;{{mvar|X}}. Then {{mvar|S}} is a subring of&nbsp;{{mvar|R}}, called the [[centralizer (ring theory)|centralizer]] (or commutant) of&nbsp;{{mvar|X}}. The center is the centralizer of the entire ring&nbsp;{{mvar|R}}. Elements or subsets of the center are said to be ''central'' in&nbsp;{{mvar|R}}; they (each individually) generate a subring of the center.

=== 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.

=== Homomorphism ===
{{main|Ring homomorphism}}
A '''[[ring homomorphism|homomorphism]]''' from a ring {{math|(''R'', +, '''⋅''')}} to a ring {{math|(''S'', ‡, ∗)}} is a function {{mvar|f}} from {{mvar|R}} to&nbsp;{{mvar|S}} that preserves the ring operations; namely, such that, for all {{math|''a''}}, {{math|''b''}} in {{mvar|R}} the following identities hold:
:<math>\begin{align}
& f(a+b) = f(a) \ddagger f(b) \\
& f(a\cdot b) = f(a)*f(b) \\
& f(1_R) = 1_S
\end{align}</math>

If one is working with {{nat|rngs}}, then the third condition is dropped.

A ring homomorphism {{mvar|f}} is said to be an '''[[isomorphism]]''' if there exists an inverse homomorphism to {{mvar|f}} (that is, a ring homomorphism that is an [[inverse function]]), or equivalently if it is [[bijection|bijective]].

Examples:
* The function that maps each integer {{mvar|x}} to its remainder modulo {{math|4}} (a number in {{math|{{mset|0, 1, 2, 3}}}}) is a homomorphism from the ring {{tmath|\Z}} to the quotient ring {{tmath|\Z/4\Z}} ("quotient ring" is defined below).
* If {{mvar|u}} is a unit element in a ring {{mvar|R}}, then <math>R \to R, x \mapsto uxu^{-1}</math> is a ring homomorphism, called an [[inner automorphism]] of {{mvar|R}}.
* Let {{mvar|R}} be a commutative ring of prime characteristic {{mvar|p}}. Then {{math|''x'' ↦ {{itco|''x''}}{{sup|''p''}}}} is a ring endomorphism of {{mvar|R}} called the [[Frobenius homomorphism]].
* The [[Galois group]] of a field extension {{math|''L'' / ''K''}} is the set of all automorphisms of {{mvar|L}} whose restrictions to {{mvar|K}} are the identity.
* For any ring {{mvar|R}}, there are a unique ring homomorphism {{tmath|\Z \mapsto R}} and a unique ring homomorphism {{math|''R'' → 0}}.
* An [[epimorphism]] (that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map {{tmath|\Z\to\Q}} is an epimorphism.
* An algebra homomorphism from a {{mvar|k}}-algebra to the [[endomorphism algebra]] of a vector space over {{mvar|k}} is called a [[algebra representation|representation of the algebra]].

Given a ring homomorphism {{math|''f'' : ''R'' → ''S''}}, the set of all elements mapped to 0 by {{mvar|f}} is called the [[kernel of a ring homomorphism|kernel]] of&nbsp;{{mvar|f}}. The kernel is a two-sided ideal of&nbsp;{{mvar|R}}. The image of&nbsp;{{mvar|f}}, on the other hand, is not always an ideal, but it is always a subring of&nbsp;{{mvar|S}}.

To give a ring homomorphism from a commutative ring {{mvar|R}} to a ring {{mvar|A}} with image contained in the center of {{mvar|A}} is the same as to give a structure of an [[associative algebra|algebra]] over {{mvar|R}} to&nbsp;{{mvar|A}} (which in particular gives a structure of an {{mvar|A}}-module).

=== Quotient ring ===
{{main|Quotient ring}}
The notion of [[quotient ring]] is analogous to the notion of a [[quotient group]].  Given a ring {{math|(''R'', +, '''⋅''')}} and a two-sided [[Ideal (ring theory)|ideal]] {{mvar|I}} of {{math|(''R'', +, '''⋅''')}}, view {{mvar|I}} as subgroup of {{math|(''R'', +)}}; then the '''quotient ring''' {{math|''R'' / ''I''}} is the set of [[coset]]s of {{mvar|I}} together with the operations
: <math>\begin{align}
& (a+I)+(b+I) = (a+b)+I, \\
& (a+I)(b+I) = (ab)+I.
\end{align}</math>
for all {{math|''a'', ''b''}} in {{mvar|R}}.  The ring {{math|''R'' / ''I''}} is also called a '''factor ring'''.

As with a quotient group, there is a canonical homomorphism {{math|''p'' : ''R'' → ''R'' / ''I''}}, given by {{math|''x'' ↦ ''x'' + ''I''}}. It is surjective and satisfies the following universal property: 
* If {{math|''f'' : ''R'' → ''S''}} is a ring homomorphism such that {{math|1=''f''(''I'') = 0}}, then there is a unique homomorphism <math>\overline{f} : R/I \to S</math> such that <math>f = \overline{f} \circ p.</math> 
For any ring homomorphism {{math|''f'' : ''R'' → ''S''}}, invoking the universal property with {{math|1=''I'' = ker ''f''}} produces a homomorphism <math>\overline{f} : R / \ker f \to S</math> that gives an isomorphism from {{math|''R'' / ker ''f''}} to the image of {{mvar|f}}.
<!-- need more work: A subset of {{math|''R''}} and the quotient {{math|''R'' / ''I''}} are related in the following way. A subset of {{mvar|R}} is called a [[system of representatives]] of {{math|''R'' / ''I''}} if no two elements in the set belong to the same coset, that is, each element in the set represents a unique coset. It is said to be complete if the restriction of {{math|''R'' → ''R'' / ''I''}} to it is surjective. -->