> [!NOTE] **Definition** (Two-, Left- & Right-Sided Ideals) > Let $(R,+,\times)$ be a [[Rings|ring]]. Let $(J,+)$ be a [[Subgroup|subgroup]] of $(R,+)$. Then $J$ is a **two-sided ideal** (or simply ideal) of $R$ iff $\forall j\in J: \forall r \in R: (j\times r \in J) \land (j \times r \in J).$ > > $J$ is a **left-sided ideal** of $R$ iff $\forall j \in J: \forall r \in R: r\times j \in J$ > >$J$ is a **right-sided ideal** of $R$ iff $\forall j \in J: \forall r\in R: j \times r \in J$ > [!Example] Examples > The [[Ideals of Integers|ideals of integers]] are the $n\mathbb{Z}$ for $n\in \mathbb{Z}.$ > > For any ring $R$, $\{0\}$ and $R$ are the trivial ideals of $R.$ > > The set of polynomials with no constant term is an ideal of $\mathbb{R}[x]$. > >The [[Unit Group of Ring|unit group a ring]] is **not** ideal (firstly, its a subgroup $(R,\times)$ not $(R,+)$). > >The [[Ring of Polynomial Forms over Field is a Principal Ideal Domain|ideals of a ring of polynomials over a field]] $F$ are the $fF[x]$ for $f\in F[x].$ # Properties Any ideal of $R$ is a [[Ideal is Subring|subring]] of $R.$ **Ideals containing the unity**: Note that [[Ideal with Unity is Whole Ring|any ideal containing the unity is the whole ring]] (i.e. if $R$ is a non-zero ring with unity then $R$ is the only ideal of $(R,+,\times)$ containing $1$). It follows that [[Ideal with Unit is Whole Ring|any ideal containing a unit is the whole ring]]. **New ideals from old**: [[Sum of Ring Ideals is Ideal]]. [[Intersection of Ring Ideals is Ideal]]. **Relation to equivalence relations**: Note that [[Congruence Relation and Ideal are Equivalent|congruence relation and ideal are equivalent]]: an ideal induces a unique congruence relation on the ring and a congruence relation on the ring induces a unique ideal of the ring. The coset space of an ideal (same as the [[Equivalence relations|quotient space]] of the ring by its congruence relation) forms a ring KA [[Quotient Ring]]. **Prime and maximal ideals**: A [[Maximal Ideal|maximal ideal]] is a maximal element of the proper ideals of $R$ ordered by the subset relation. A [[Prime Ideal|prime ideal]] is ...