Editing Ring (mathematics) (section)

=== 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.