Integrally closed domain

Imagine you have a special kind of box called a "domain." This box is like a special club where only certain kinds of numbers can be inside. Let's say the numbers inside this box are all fractions, like 1/2 or 3/4.

Now, if you have another box called an "integrally closed domain," it's like a fancier version of the first box. This special club only allows certain kinds of numbers inside as well, but they have to be really special. They have to be numbers that can't be broken down into smaller, simpler fractions.

For example, imagine the number 1/3. This wouldn't be allowed inside the integrally closed domain, because we can simplify it down to 2/6 or 3/9 or many other fractions. But a number like 2/5 can't be simplified any further, so it would be allowed inside the integrally closed domain.

In other words, an integrally closed domain is like a special club where the numbers inside are fancy and can't be simplified any further.