structural:reformatting
Generalizations

Ideals can be generalized to any monoid object (R, ⊗) {\displaystyle (R,\otimes)}, where R {\displaystyle R} is the object where the monoid structure has been forgotten. A left ideal of R {\displaystyle R} is a subobject I {\displaystyle I} that "absorbs multiplication from the left by elements of R {\displaystyle R} "; that is, I {\displaystyle I} is a left ideal if it satisfies the following two conditions:

A right ideal is defined with the condition " r ⊗ x ∈ (I, ⊗) {\displaystyle r\otimes x\in (I,\otimes)} " replaced by "' x ⊗ r ∈ (I, ⊗) {\displaystyle x\otimes r\in (I,\otimes)} ". A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. When R {\displaystyle R} is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.

An ideal can also be thought of as a specific type of R -module. If we consider R {\displaystyle R} as a left R {\displaystyle R} -module (by left multiplication), then a left ideal I {\displaystyle I} is really just a left sub-module of R {\displaystyle R}. In other words, I {\displaystyle I} is a left (right) ideal of R {\displaystyle R} if and only if it is a left (right) R {\displaystyle R} -module that is a subset of R {\displaystyle R}. I {\displaystyle I} is a two-sided ideal if it is a sub- R {\displaystyle R} -bimodule of R {\displaystyle R}.

Example: If we let R = Z {\displaystyle R=\mathbb {Z} }, an ideal of Z {\displaystyle \mathbb {Z} } is an abelian group that is a subset of Z {\displaystyle \mathbb {Z} }, i.e. m Z {\displaystyle m\mathbb {Z} } for some m ∈ Z {\displaystyle m\in \mathbb {Z} }. So these give all the ideals of Z {\displaystyle \mathbb {Z} }.