Editing Ideal (ring theory) (section)

== Definitions ==
Given a [[ring (mathematics)|ring]] {{mvar|R}}, a '''left ideal''' is a subset {{mvar|I}} of {{mvar|R}} that is a [[subgroup]] of the [[additive group]] of <math>R</math> that "absorbs multiplication from the left by elements of {{tmath|1= R }}"; that is, <math>I</math> is a left ideal if it satisfies the following two conditions:
# <math>(I,+)</math> is a [[subgroup]] of {{tmath|1= (R,+) }},
# For every <math>r \in R</math> and every {{tmath|1= x \in I }}, the product <math>r x</math> is in {{tmath|1= I }}.<ref>{{harvnb|Dummit|Foote|2004|p=242}}</ref>

In other words, a left ideal is a left [[submodule]] of {{mvar|R}}, considered as a [[left module]] over itself.<ref>{{harvnb|Dummit|Foote|2004|loc=§ 10.1., Examples (1).}}</ref>

A '''right ideal''' is defined similarly, with the condition <math>rx\in I</math> replaced by {{tmath|1= xr\in I }}. A '''two-sided ideal''' is a left ideal that is also a right ideal.

If the ring is [[commutative ring|commutative]], the three definitions are the same, and one talks simply of an '''ideal'''. In the non-commutative case, "ideal" is often used instead of "two-sided ideal".

If {{mvar|I}} is a left, right or two-sided ideal, the relation <math>x \sim y</math> if and only if
:<math>x-y\in I</math>
is an [[equivalence relation]] on {{mvar|R}}, and the set of [[equivalence class]]es forms a left, right or bi module denoted <math>R/I</math> and called the ''[[quotient module|quotient]]'' of {{mvar|R}} by {{mvar|I}}.<ref>{{harvnb|Dummit|Foote|2004|loc=§ 10.1., Proposition 3.}}</ref> (It is an instance of a [[congruence relation]] and is a generalization of [[modular arithmetic]].)

If the ideal {{mvar|I}} is two-sided, <math>R/I</math> is a ring,<ref>{{harvnb|Dummit|Foote|2004|loc=Ch. 7, Proposition 6.}}</ref> and the function 
:<math>R\to R/I</math> 
that associates to each element of {{mvar|R}} its equivalence class is a [[surjective]] [[ring homomorphism]] that has the ideal as its [[kernel (algebra)|kernel]].<ref>{{harvnb|Dummit|Foote|2004|loc=Ch. 7, Theorem 7.}}</ref> Conversely, the kernel of a ring homomorphism is a two-sided ideal. Therefore, ''the two-sided ideals are exactly the kernels of ring homomorphisms.''

=== Note on convention ===
By convention, a ring has the multiplicative identity. But some authors do not require a ring to have the multiplicative identity; i.e., for them, a ring is a [[rng (mathematics)|rng]]. For a rng {{mvar|R}}, a '''left ideal''' {{mvar|I}} is a {{not a typo|subrng}} with the additional property that <math>rx</math> is in {{mvar|I}} for every <math>r \in R</math> and every <math>x \in I</math>. (Right and two-sided ideals are defined similarly.) For a ring, an ideal {{mvar|I}} (say a left ideal) is rarely a subring; since a subring shares the same multiplicative identity with the ambient ring {{mvar|R}}, if {{mvar|I}} were a subring, for every <math>r \in R</math>, we have <math>r = r 1 \in I;</math> i.e., <math>I = R</math>.

The notion of an ideal does not involve associativity; thus, an ideal is also defined for [[non-associative ring]]s (often without the multiplicative identity) such as a [[Lie algebra]].