📂Abstract Algebra

Ideals in Abstract Algebra

Definition¹

A subgroup $(I, +)$ of a ring $(R , + , \cdot )$ satisfying $a I \subset I$ and $I b \subset I$ for all $a,b \in R$ is called an ideal.

Explanation

As a simple example, $n \mathbb{Z}$ is an ideal of $\mathbb{Z}$. The name ideal comes, quite literally, from the word ideal. It is called that precisely because it is an ideal subgroup to work with in abstract algebra.

In particular, if $R$ is a commutative ring, then in the sense that $I$ becomes a normal subgroup of $R$, one simply writes $I \triangleleft R$. Just as normal subgroups were important in group theory, we can guess that ideals will appear prominently in all sorts of theorems of ring theory. The reason we say ring theory in particular is that the ideal is essentially a concept exclusive to rings.

An ideal $I$ is a subring of $R$.

In the definition, to emphasize the contrast with groups, we said a 'subgroup' satisfying the conditions, but in fact it naturally becomes a subring as well. We will not go so far as to prove it, but if it is hard to grasp, just think carefully about the conditions $a I \subset I$ and $I b \subset I$. Intuitively, $I$ is a 'set of elements that held out', remaining an algebraic structure even 'when multiplication is applied' by every element of $R$. Common sense suggests that $(I , \cdot )$ constructed this way should at least manage to be a semigroup with respect to $(R , + , \cdot)$. Of course, this explanation is not mathematical, so if you are really doubtful, verify it directly using the subring test. In fact, depending on the textbook, it is sometimes defined as a subring from the very start.