Ideals of Rings

# Ideals of Rings

 Definition: Let $(R, +, *)$ be a ring. An Ideal (or Two-sided Ideal) is a subset $I \subseteq R$ such that: 1) $(I, +)$ is a subgroup of $(R, +)$. 2) For all $r \in R$ and for all $i \in I$ we have that $r * i \in I$ and $i * r \in I$, i.e., for all $r \in R$, $rI \subseteq I$ and $Ir \subseteq I$.

Some others distinguish a difference between left ideals (defined as subrings for which we only require that for all $r \in R$, $rI \subseteq I$), and right ideals (defined as subrings for which we only require that for all $r \in R$], $IR \subseteq I$.

If $(R, +, *)$ is a ring and if $0$ is the identity for $+$ then $\{ 0 \}$ is an ideal (the trivial ideal), and the whole set, $R$ is also an ideal.

Recall from the Normal Subgroups page that if $(G, *)$ is a group and $(H, *)$ is a subgroup then $(H, *)$ is said to be a normal subgroup if for all $g \in G$ we have that $gH = Hg$ (the left coset of $H$ with representative $g$ equals the right coset of $H$ with representative $g$), and given a normal subgroup $(H, *)$ of $(G, *)$ we can construct the quotient group $(G/H, \cdot)$ where $G / H$ is the set of all left cosets (which of course is also the set of all right cosets), and $\cdot$ is the operation on $G / H$ defined for all $g_1H, g_2H \in G/H$ by:

(1)
\begin{align} \quad (g_1H) \cdot (g_2H) = (g_1 * g_2)H \end{align}

The concept of an ideal subring will hold the spot of a normal subgroup when we generalize quotient groups by quotient rings.

We will now classify some very important types of ideals.

 Definition: Let $(R, +, *)$ be a commutative ring and let $a \in R$. The Principle Ideal of $a$ is the ideal $= \{ a * r : r \in R \}$.

We require that $(R, +, *)$ is a commutative ring so that $a * r = r * a$ for a fixed $a$ and for all $r \in R$.

 Definition: Let $(R, +, *)$ be a ring and let $(P, +, *)$ be an ideal with $P \neq \{ 0 \}$ ($0$ being the identity for $+$) and $P \neq R$. Then $(P, +, *)$ is a Prime Ideal if for all $a * b \in P$ we have that $a \in P$ or $b \in P$.
 Definition: Let $(R, +, *)$ be a ring and let $(M, +, *)$ be an ideal. Then $(M, +, *)$ is a Maximal Ideal if no other ideal of $(R, +, *)$ (except the whole ring) contains $(M, +, *)$.