Ideal, integral domain, PID, prime ideal

Question

I am very confused with above terms, can anyone please show me examples and/or a good way to remember and understand their differences? Thank you!

Answer 1

An integral domain is a commutative ring $R$ such that if $ab = 0$, then either $a = 0$ or $b = 0$ for all $a,b \in R$. This is equivalent to the statement that $0$ is the only zero-divisor. This is not true in general - consider $\mathbb{Z}/24\mathbb{Z}$. Neither $4$, nor $6$ are $0$ as elements of $\mathbb{Z}/24\mathbb{Z}$, yet $6 \cdot 4 = 24 = 0 \in \mathbb{Z}/24\mathbb{Z}$.

A principal ideal domain (PID) is an integral domain in which every ideal is principal. (I define what it means to be a principal ideal below.)

An ideal $I$ is a subgroup of the additive group of a ring $R$ that is closed under multiplication with respect to all the elements of the ring. In symbols, $\forall r \in R$ $rI \subseteq I$. For an example, let $R = \mathbb{Z}$ and consider $d\mathbb{Z}$ for any $d \in \mathbb{Z}$. This is an ideal of $\mathbb{Z}$.

(Two-sided) ideals are important because they are precisely the subsets of a ring $R$ that you can use to quotient the ring to get a well-defined quotient ring $R/I$. This is similar to the idea of quotient groups (which you should be familiar with.)

There are subtleties here - most people only ever deal with two-sided ideals, but ideals can also be exclusively left-sided or right-sided. You can only factor rings with two-sided ideals. You can look up the difference on Wikipedia.

An ideal $I$ of a ring $R$ is said to be prime if $ab \in I$ implies that either $a \in I$ or $b \in I$. Equivalently, $R/I$ is an integral domain.

An ideal $I$ of a ring $R$ is said to be principal if $I = aR$ for some $a \in R$.

I hope that helps clarify the difference.