Definition of a Ring Part 3

welcome back to this next video which we are discussing the definition of a ring okay so we've now seen the definition of a ring we've seen a ring is a set of symbols fundamentally with two composition laws defined on it addition and multiplication addition must obey the axioms of an abelian group and multiplication must obey closure associativity have a multiplicative identity and the left and right distributive laws okay what we're now going to do is explore in a little bit more detail some of the consequences of left and right distributivity and the first consequence that we're going to look at is what do the column and row corresponding to the additive identity on them sorry you can't see this what to the column and row corresponding to the additive identity 0 in the ring look like in the multiplication table ok so we're going to a question what is a 0 times some other elements of the Ring and what is some element of the ring times 0 ok right what we can use left and right distributivity to gain some understanding of this ok so we'll start off with left distributivity here ok so this fact here left distributivity this is true no matter what x the matter what Y and the matter of what the stands you pick from the ring so I'm gonna keep x and y completely general but I'm gonna substitute in for Zed the additive identity and see what we get here so what I'm going to do is I'm going to get x times y plus the additive identity it's a perfectly good element of the Ring okay it's an element of the Ring that will always be there and the left distributive law here tells me that this will be x times y plus x times 0 which I'll just write by claps ok so that has to be true the instant we insist on left distributivity this has to be true but we can do more here ok because y plus the additive identity what's that going to equal well we never you take any element of the ring and add it to the additive identity you get same elements of the ring back again so this is just going to become x times y so I've got the x times y is equal to x times y plus x times 0 now what I can do is add on to both sides of this equation the additive identity sorry the additive inverse of x times y so add on the additive inverse of x times y x times y after all is just some elements of the ring okay all elements of the ring have an additive inverse so just add this on to both sides okay on this side we'll get x times y plus it's additive inverse which will give me the zero elements of the ring the additive identity 0 in my ring okay just by definition and on the other side when we add the additive inverse of x times y on it will cancel with this to give again the additive identity and the additive identity Plus this will just equal this so what we now get is that x times 0 is equal to 0 and that was completely general I never set any condition on X whatsoever so I've now got the fact that any element of the ring times 0 is equal to 0 now what does that mean well that tells me about the column corresponding to 0 in the multiplication composition table ok so going back up to this picture here I can take any element of the ring here any little X I like from the ring again that corresponds to taking the row corresponding to little X and then I can multiply this with zero so intersector with the column that corresponds to 0 and what that will always give me is 0 and that works no matter what little excess so that tells me that this entire column corresponding to the additive identity in the multiplication Composition table is just full of zeros it's just a column of zeros all of the answers in those boxes are just the additive identity of the ring 0 back again ok now to gain insight then into the row corresponding to the additive identity I'm gonna use right distributivity now okay so the exact same little trick here but now use right distributivity so once again I'm going to say okay let's let Z equal the additive identity applying right distributivity ban y plus zero times X is going to equal Y times X plus zero times X okay just applying this exact rule but I know I've insisted now it's going to be true for our ring okay but again I can say that Y plus zero is going to equal Y so on this side we just get Y times X add the additive inverse of Y times X onto both sides and we then get that zero times X is equal to 0 again so 0 times any element of the ring is going to give 0 back again and that now tells me about the row corresponding to 0 in the multiplication composition table it says that if I take the row corresponding to 0 and intersect that with the column corresponding to any element little X of a ring then the answer here in this entry of the multiplication table is just going to be 0 okay and that's true no matter what little X you take from the ring so this entire row then it's just going to be lower than those as zeros and I apologize that I've got two zeros there and only one there okay that's a little bit inconsistent but you get the picture and okay this is just a row for those zeros all of the answers are just zeros okay so 0 multiplied by any other element of the Ring either way round to give 0 absolutely always okay so the instant you insists the left and right distributivity are true for a ring and which we do insist then you automatically have to a set be multiplicative so that the additive identity are all multiplied by any other elements of the ring to give back the additive identity 0 multiplies back any other elements of the room to give steyr aback again ok so that's the first consequence then of distributivity being true next consequence what I want to see now is that we can extend distributivity okay so the first thing I'm gonna consider is what if we have something of this form x times and now I'm gonna have a whole bunch of elements of the ring added together and a finite number of them so we'll have y 1 plus y 2 plus and let's go this time all the way up to Y n okay so all these Y eyes these are going to be elements of the Ring and X is going to be an element of the ring as well so all of these things here are elements of the Ring okay so I've just got a finite number of elements of the ring added together like so that will give me some other elements of the ring and then I'm going to multiply this by X okay now the intuitive thing that beautiful nice thing that we'd like this to be the intuitive answer is that we'd like to expand this in the way that we learn in classical algebra 2x times y 1 plus x times y 2 plus all the way along to x times y n okay so that's what we like to do but why can we do that this is the right answer this thing here is going to equal this but I need to give you the justification for why we can do that why we can extend distributivity like this why can we stretch distributivity out because all I've insisted so far is that if you have two elements of the ring added together and you then multiply by X on the left that this applies I haven't said for an arbitrary number of finite an arbitrary finite number of things in the ring added together that it will apply okay so I need to give you the arguments as to why the instant this is true it also implies that this is true okay so the reason is that what you can do is you can consider this is just being two things added together and the way you can do that is you can consider y1 there's one thing in the ring and then you can consider all the rest y2 plus y3 plus y4 all the way up to plus yn you can consider that another element of the ring then you can apply the less distributive law to this and break it into x times y one plus x times the rest okay so y2 plus y3 plus all the way up to Y n like so again that's just applying the left distributive law that we insisted was true where we've got some multiplication and then distributing over addition of two elements of the Ring okay so here's one element of the Ring and here's the other element of the Ring okay but then of course what we can do is apply the same trick again we can say this thing here we've got y2 plus y3 all the way up to yn I can say let's consider y1 as one thing in the ring and that's consider the all the other things added together so y3 plus y4 plus all the way up to Y and there's another element of the ring and then again I can apply my left distributive law to say that this is x times y 2 plus x times the rest here so y3 plus all the way up to Y N and this can just go on them on you can then apply the same thing to this until you eventually break this down into this thing that I've got written up here okay so because there's a finite number of things in this sum here it will eventually come to an end and you will have written it out in this decomposed form here okay so distributivity then can be extended in this way where you've now got a finite number of elements of the Ring added together ok and of course this argument works just as well from right to distributivity ok so you can extend up the number of things that you've got added together here ok and it will still work ok the exact same proof follows so putting the two together then now let's consider and I'll just have to get another piece of paper for this let's consider something of this form ok like so so we'll have something of this form x1 plus x2 plus all the way up to X times y 1 plus y 2 plus all the way up to Y n ok can we apply the familiar distributivity of classical algebra to this can we multiply this out in our ring in the way that we would guess we could from classical algebra and indeed we can I'll just state rigorously all these x i's and y JS these are all elements of the ring capital r okay right so what this means then is add all of these together so add X 1 to X 2 all the way up to X and in the ring and then multiply that by all of these things added together so add y1 to y2 alright to yn get some answer in the ring and then add them together ok and obviously what we'd like to split this down into is this one multiplied by this 1 plus this 1 multiplied by this 1 plus etc etc and then we move on to the next one etc like so ok so how can we do this well the reason that we can do this is we can view this or there's just one element of the ring so initially just view this X 1 plus X 2 all the way up to X they have just view that as an element of the Ring if you like you can write replace it with an X here just view that as being one element of the Ring then we've just got an element of the Ring multiplied by a whole bunch of things added together here and now I just reduced it back to the case that we've just done I've reduced it back to this case here where we've got X multiplied by Y 1 plus y 2 plus all the way up to Y n ok so I can apply the result that we've just got like so so what we'll get is this entire thing X 1 plus X 2 plus all the way up to X n multiplying Y 1 and then we'll have plus and then we'll have X 1 plus X 2 plus all the way up to X and multiplying Y 2 and then we'll continue on plus dot dot dot all the way up to we'll have X 1 plus X 2 plus all the way along to X n multiplying Y n like so okay so that's justifying the result that we've got previously and now what we can do is we can apply the homologous result for right distributivity to each of these things here okay so we can say now let's split this up okay and again as I argued the exact same thing holds for right distributivity but you can have an arbitrary finite string of elements of the ring being added together here and it will break down in the way that you would think it's natural for it to break down okay so let's do this one here so we'll get X 1 times y 1 plus X 2 times y 1 plus and it will continue on all the way up to X n times y 1 okay so that's the first line here then we'll get the second line plus X 1 times y 2 plus X 2 times y 2 plus all the way along to X n times y 2 and then you'll go down all the way along to the final line here just bring this up a little bit and we'll have then X 1 times yn plus X 2 times yn plus all the way along to X M times yn okay so indeed as soon as right enough distributivity are true in a ring we can expand things of this form in the common sense way that you'd think would be the way that it should be expanded from classical algebra okay so right and left distributivity can be extended to this result here okay right so what I now wanted to is discuss a few other pieces of terminology okay so the first piece of terminology that I want to discuss is the concept of a unit in the ring okay so you will hear the term unit all the time when you're working in rings okay so what is meant by a unit in a ring okay so an element of a ring is called a unit if it has a multiplicative inverse in the ring okay so we saw that in the definition of a ring nowhere did we insist that elements the ring must have multiplicative inverses okay that was never an axiom that multiplication had to obey however it might just happen that some elements of the ring do have multiplicative inverses and if they do have multiplicative inverses then those elements are called units okay so here's the definition of a unit so a unit if so let's say firstly let's just have X is an element of the ring X is going to be called a unit so X is a unit if there exists another element of the ring which I'll call one over X the multiplicative inverse of X you can always so use the notation X to the power of negative one and this needs to be an element of the ring such that x times one over X is equal to the multiplicative identity 1 and also the other way around so 1 over X times X is equal to 1 okay so if you multiply these two together both ways round it gives the multiplicative identity one back again so an element of the ring is called a unit of the ring if it has a multiplicative inverse in the ring again the multiplicative inverse is something that satisfies both of these equations I if you multiply them both ways round you get the multiplicative identity for back again okay now if an element of the ring does have a multiplicative inverse it will only ever have one multiplicative inverse then the reason that that is true is exactly the same reason as we gave for y elements of the ring can only have one additive inverse and we insist the multiplication all obeys associativity and the instant you insist on that any element of the ring cannot have more than one multiplicative inverse otherwise it would violate associativity in exactly the same way as having more than one additive inverse with an violated additive associativity okay so the exact proof holds true for multiplication just change the symbols of it a bit change the additional symbols to multiplication symbols and the exact same proof holds true okay so if an element of the ring has a multiple negative inverse at all it will only have one multiplicative inverse otherwise it would violate associativity and hence we wouldn't be talking about ring theory okay so that's the definition then of a unit okay one final definition that I now want to give you okay we can extend a ring there is a concept that is beyond the general ring okay and the next concept is the concept of a commutative ring okay similar to the concept of an abelian group so we're now going to define what is meant by commutative ring now in this first video I wanted to give you the truly general definition of a ring and I have now done that okay the definition that we've just looked at is the formal definition of a general ring however in the rest of this playlist on ring theory we are not going to be studying ring theory in its full generality we are going to be studying commutative rings okay because those are the slightly more important ones normative rings are they they beautiful and very very fascinating but they're not quite as important as commutative rings okay all the while the most important example of a ring the integers which we'll come onto in a moment that is a commutative ring is a commutative ring theory is arguably more important than non commutative ring very okay so what then is the definition of a commutative ring well you take the exact definition of a ring and you insist on one more axiom okay so if I just get like axioms of ring theory back up again addition doesn't change at all addition is still going to be an abelian group okay we're just going to add an additional axiom on to the axioms that the multiplication must have been I mean you might be able to guess what axiom we're going to add on we're going to add on that multiplication must be commutative okay so I'm going to have axiom number five here that the multiplication composition table must obey again it's going to say you pick any two elements of the Ring so for all x and y that you can possibly pick in the ring capital R it must be the case that x times y is equal to y times X okay so it does not matter which way around you multiply two elements of the ring together the answer will always be the same okay and again if I draw out the picture of what that means from multiplication table okay so if here is our multiplication table for our ring which I'll color it in red here remember that these two different ways of doing it are not the same entry in the multiplication table and that's of course X and Y is equal to the same thing in the ring which in which would mean that these two were the same thing in the composition table but x times y more generally will look like this so this is the row dedicated to little X and then you'll take the column dedicated to little Y then of course the inception of those two things here this will have the answer to x times y and then if you want to do it the other way around you'll take the row dedicate it to little Y okay so here is the road at a case to the Y you'll take the column dedicated to little X here and the answer to Y times X will then be this entry in the composition table so again you can see that if you were just to make up the entries in this composition table it might not be the case that these two would end up having the same answer in them and therefore your multiplication and composition law wouldn't be commutative okay so it's not completely trivial that this is true and what it is actually going to correspond to meaning is that the multiplication table must be symmetric down this diagonal line here so all of the entries in the multiplication table must be the same as the mirror image entry okay right so in a commutative ring we're going to insist that multiplication also obeys commutativity on top of all the other axioms that we insisted it must obey again throughout this playlist we are going to concentrate completely on commutative rings okay now I might at some points in future videos forget to add the word commutative ring in front of room so I might just say that's our beer ring whenever I say ringing from now on I will mean commutative ring I will try to remember to say commutative ring but if I do forget just as soon I mean commutative ring we're not going to touch long commutative rings at all long commutative rings is all about matrices okay because you can add matrices together and you can multiply them and matrix multiplication isn't necessarily commutative but it does AB a the other axioms that we need it to obey such as distributivity most significantly and associativity is also a significant one okay right so that learn is the definition of a commutative ring I think what would you now is have a break and in the next video what we'll do is we'll look at some examples of rings we'll look at the prototypical example of the integers okay then we'll look at the fact that all fields that we've seen in previous playlist specifically the playlist on vector spaces are rings and then we'll talk about the zero room for a bit and then we'll leave by finally defining what a suffering is

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