Algebra, Group, Ring, Rng, Field, Monoid, Vector space | Abstract algebra systematized

0:00 When I started my first year at university, the very first algebra lectures left me a bit confused. They introduced various algebraic structures without the slightest explanation of why I needed them. I ended up using some of them only 10 years later. Honestly, I could never remember which one was which. Every time I had to Google it again, and that was also not easy. There are tons of pictures which try to connect all these structures but none of them were exactly what I wanted. So I just tried to systemize them on my own. You will find the link to the PDF file in the description. But before you download it, help me a little bit. I hate to explain this, but the point is that my YouTube channel is incredibly unpopular and

0:46 my videos don't even get watched the amount of hours I wasted to make them. They're probably not perfect. I'm a scientist, not an actor, but I'm really doing my best. So, if you find this useful, please watch this video till the end, put a like or subscribe. Probably after your like, YouTube will finally start recommending it to the target audience and give it a chance to at least be Googleable by the people who really need it, who are looking for this information. And of course, I'm not paid for these videos. I'm doing them in my free time. Thank you for watching, and let's move to the main part. Let's now discuss how to use this diagram. First of all, everything is about sets, or informally speaking, we have a collection of elements, any elements

1:37 like this. All the objects mentioned here are just sets with some additional properties. These properties are called axioms and they are listed in the center of the diagram. Different sets may satisfy different amount of these properties. So how do we read this diagram? Latin letter X Y etc. and denote elements of a set. In the center, we are talking about binary operations like addition and multiplication. Every item is an axiom. The pink rectangle show which axioms are satisfied for a particular structure. For example, group or algebra. In the rectangle, we can also find some examples of these structures. But we need to keep in mind that addition doesn't necessarily mean addition in the usual sense. Same for multiplication.

2:26 The plus and multiplication symbols are just that ---symbols. They can represent any operation. What matters is not the name of the operation but its properties. For instance, addition might actually be multiplication or subtraction in the usual sense. If that's how it works in for this particular set, it's very important to understand that at this point we know absolutely nothing except for these axioms. We don't know what plus means or what times means. And the best part, we don't even know what the scalar is even though it's mentioned in some of the axioms. So for now, we will ignore the central part, axioms 5 through 9, and talk about it later. Let's take the integers as an example and find out which axioms they satisfy. And for this specific example,

3:16 we assume that our symbols plus and multiply have the usual meaning. It's easy to check that they satisfy all the usual axioms for addition. Zero is the identity and the inverse of the element x is just minus x. Multiplication satisfies most axioms as well. But there is a problem with the multiplicative inverse. Most integers do not have one within the set of integers. For example, the inverse of two would be 1/2 which is not an integer. So the integers form a commutative ring with unity but not a field. And it is completely okay that they are both commutative ring and ring with unity because if I would add all these pair cases into the diagram it would be larger. But in

4:07 reality a lot of combinations are possible. So why do we even care about axioms? Let's take a simple example. The equation b x + a= c. We just solve it. X = (C - A)/ B. Easy. But hold on. What kind of numbers we are working with? If we are in the world of rational numbers, no problem. We can divide by b and the solution is always there. But if we are working with integers, that's not always true. Say a = 1, c = 0, b = 2, then x = -1 / 2, which isn't an integer. So in the integers, the

4:52 equation might not have a solution. That's where axioms come in. They tell us what operations are allowed and what properties we can rely on. For example, if we are in a field, we know we can add, subtract, multiply, and divide except by zero. There is always an inverse for non-zero elements and everything behaves just nicely. So in a field b x + a= c always has a solution because dividing by b is guaranteed. But in the integers we don't have division. So equations just don't work out. In other words, axioms are like the rules of the game. If you want to prove something, we have to know what moves you allowed to make. Otherwise, you're just guessing. Let's now consider something

5:41 wild, unusual addition. So, sometimes addition and multiplication are nothing like what we're used to. Example, let's take the set of real numbers and define a new addition operation as x + y equals x * y in usual sense minus x also in usual sense. Sounds a bit weird, doesn't it? But it is just the way how we defined this plus sign. Let's check whether this operation satisfies the first axioms. Associativity. It's a little bit tricky. Let's substitute everything and for the left hand side we get this expression and for the right hand side we get this expression

6:28 which are definitely not equal to each other. So the operation is not associative. Now let's check identity element. So we want to find an element which behaves as zero under addition, under this strange addition. So let's check it. And we see that there is no identity element. We cannot do this. This a also fails. Let's check inverse element. We don't need to check this because identity doesn't exist and commutativity is obviously not satisfied. So basically only closure is satisfied and such a structure is called a magma. If the first three axioms are satisfied, it is called a group. If it is also commutative, it is abelian group, the pink rectangles just show

7:17 which axioms are satisfied. If every axiom for both addition and multiplication is satisfied, we have a field. If there are some troubles with multiplication, then it's a ring. And now we move on to axioms 5 to 9. If we have already defined an algebraic structure K which can be a ring, a field or even something else, we say that the Greek letters alpha and beta denote elements of this ring or field K. We call them scalars here. And now we can finally multiply X and Y which will be now the elements of another algebraic structure say L by alpha and beta because we have defined what alpha and beta are. And then we say that we have some structure L over K which means for

8:06 example a vector space L over the field R. And that means that for this particular structure which satisfies axioms 5 till 9 alpha and beta belong to the field K. In in other words to define a structure where our objects x y that etc live, we need to fix another basic algebraic structure K. This basic structure K consist of scalars for which we use Greek letters alpha and beta. It sounds quite tricky but when you start really work with, for example, vector spaces it will be quite intuitive and you will not remember all the axioms all the time, but it's useful to keep in mind that they exist. And some discussion. As I told there are examples of each structure in the diagram.

8:56 Don't be scared if you're not familiar with them. I really tried to find examples that satisfy exactly these axioms and not more, but I might have made a mistake. If you notice any mistakes or inaccuracies, please write in the comments and I will update the file. Not all possible algebraic structures are listed here, otherwise the table would be too big. But of course, we can always say something like commutative and associative algebra if both properties are satisfied, or unital commutative ring with division. Sometimes it's also useful to view structures that satisfy many axioms as simpler objects. For example, if in a specific problem we only need to add real numbers, we can ignore the fact that real numbers also support multiplication

9:47 and consider them as a group, for example. Thank you for watching and don't forget to subscribe.