Injective hull

Imagine you have a big box that can hold all of your toys. Now, imagine you have another smaller box that can fit inside the big box. You can put your small box of toys inside the big box and there will still be room for more toys. But what if you want to put a toy in the small box that won't fit? You can't just squish it in, it won't fit.

In math, we have a similar concept called the injective hull. We start with a small set of numbers, called a module. We want to add more numbers to it but we don't want to change what we already have. We create a bigger module, called an injective module, that can hold our small module and more. We can put all our numbers from the small module in the injective module, but we may also add more numbers that don't fit in the small module.

The injective hull is the smallest injective module that contains our small module. It's like finding the smallest big box that can hold our small box and more toys but won't change our small box of toys.