Semigroup ring

Okay kiddo, let's talk about semigroup rings.

So imagine you have a group of friends and you all have one toy each. You can play with your toy by yourself, but you can also combine your toys with your friends' toys to make bigger and better toys.

That's sort of like what a semigroup is. It's a group of things that you can combine to make bigger and better things. However, in a semigroup, you can't undo the combining or take anything away.

Now, imagine that instead of toys, each friend has a number. You can still combine them to make bigger and better things like before, but now you can do arithmetic with them too.

That's sort of like a semigroup ring. It's a way of combining these numbers (or objects) to make bigger and better things, but now you can also do arithmetic with them. You can add them together or multiply them, just like numbers.

But there's a twist. Remember how we said you can't undo the combining or take anything away in a semigroup? Well, in a semigroup ring, you can sort of undo the combining or take things away, but only if you add in some extra rules to make it work.

It's like playing with your toys again. You can take them apart and put them back together, but only if you follow the rules. Maybe you have to leave a certain toy on the bottom or you can only combine certain toys together.

In a semigroup ring, these extra rules are called axioms. They help make sure that the arithmetic works properly, even when you're undoing combining or taking things away.

So in summary, a semigroup ring is like a group of friends with numbers instead of toys. You can combine them to make bigger and better things and do arithmetic with them, but you have to follow some extra rules so that it works properly.