And also, another thing, I'm curious about. They say that an Ideal is the analogue of the Normal subgroup in group theory, but that confuses me.
Let a Group be G. Let a subgroup be H. H is normal in G if and only if, $gH = Hg$, for all $g$ in $G$.
Let $<R,+,*>$ Be a ring with set $R$ and the usual operations of addition and subtraction. Let $I$ be a subset of $R$. We say that $I$ is an ideal if it forms a subgroup $<I,+>$ of $<R,+>$ under addition. And also, for $a,b$ that is in $I$:
$a*b$ is in $I$ and $b*a$ is in $I$.
So a subgroup $H$ is normal if and only if for all elements $aH = Ha$, but a SET is an ideal if the following conditions hold. I don't understand the analogue of the two. An ideal requires that multiplication be defined and in our set I. Normal subgroups require something different. What am I missing?