📂Abstract Algebra

Ideals in Abstract Algebra

Definition ¹

A subring $(I, +)$ that satisfies $a I \subset I$ and $I b \subset I$ for all $a,b \in R$ in a ring $(R , + , \cdot )$ is called an Ideal.

Explanation

As a simple example, $n \mathbb{Z}$ is an Ideal of $\mathbb{Z}$. The name Ideal literally comes from the word Ideal, as it is the perfect subring to deal with in abstract algebra.

Especially if $R$ is a commutative ring, $I$ being a normal subgroup of $R$, it’s also commonly referred to just as $I \triangleleft R$. Just like how normal subgroups are important in group theory, ideals will play a significant role in various theorems of ring theory. The reason it’s specifically called ring theory is because Ideal is essentially a concept unique to rings.

The Ideal $I$ is a subring of $R$.

Although the definition emphasizes the comparison with groups by mentioning a ‘subgroup’ that meets certain conditions, it naturally also becomes a subring. No proof will be provided here, but if it’s hard to understand, think thoroughly about the conditions $a I \subset I$ and $I b \subset I$. Seemingly, $I$ is a collection of elements that ‘survive under multiplication’ by any element of $R$, maintaining its structure as an algebraic structure. Logically, such a construction like $(I , \cdot )$ should at least form a semigroup with respect to $(R , + , \cdot)$. This explanation is not mathematical, so if you’re still doubtful, verify it directly using the subring test. In fact, some textbooks might even start with defining it as a subring from the beginning.