 Welcome back to our lecture series Math 4230 abstract algebra 2 for students at Southern Utah University. As usual it'll be your professor today Dr. Andrew Missildine. In lecture 14 we want to talk some more about the idea of an ideal, right? What was an ideal just as a reminder? So the ideals for rings are just like normal subgroups for groups, okay? An ideal is of course a subring, right? So if we have some ideal we say it's a subring of the ring and we use the same normality symbol here to suggest that this isn't just a subring it's an ideal, right? What's important about an ideal? Well an ideal is closed under addition, right? So if we take any sum of two elements and so this is here a set containment. If you take an element of the ideal plus another element you get back in there. So in particular it's an additive subgroup but it also has this multiplicative closure that transcends just i times i is a subset of i. We can do better. In fact we can get that for all elements r inside the ring. We're gonna get that r i and i r is a subset of i like so. So you get closure in a stronger sense that if you multiply an element of an ideal by any element of the ring you do belong to the ideal still. Now the big thing about ideals is ideals coincide with kernels of ring homomorphisms, okay? And as such we can construct quotient rings using ideals and the theory of ideals carries over pretty much exactly the same to rings as it did for groups. In particular in this video I want to summarize the three isomorphism theorems for ring theory and their proofs are by analog to what we saw with group theory. So the first one the first isomorphism theorem sometimes called the fundamental homomorphism theorem says that if you have a ring homomorphism we'll call it psi as a map between two rings r and s. This is a ring homomorphism and suppose the kernel of psi is this ideal i. We've seen previously that the kernel of a ring homomorphism is exactly an ideal. Ideals and kernels are the same things. Every ideal there is the natural homomorphism attached to it but also every ideal of a homomorphism is in fact an ideal as well. So kernels and ideals are the same thing for ring theory just like normal subgroups and kernels are the same thing in group theory. Alright so if we have this ring homomorphism psi is a map from r to s and the kernel of psi is equal to i then there exists a unique homomorphism this time we'll call the map phi. It goes from the quotient ring r mod i to the ring s and be aware that psi is then going to be the composition of two homomorphisms. There's this homomorphism phi and then there's a homomorphism eta here where eta is then the natural map just like with just like with groups. There's this natural quotient map that goes from a ring r to its quotient ring r mod i which be aware here eta of r just sends r to the ideal that's represented by r so that's the natural map in this sense. Okay so then with that with that in mind we then have in particular phi you know so the same in the first isomorphism is guaranteeing that there's these unique homomorphism we can always fit that homomorphism there in particular if you take the map phi that was constructed here this will form an isomorphism between r mod i with the image of psi because psi might not be an onto map but if it is we're going to well whatever the image turns out to be onto or not the phi forms an isomorphism between r mod i and the image of psi so in particular much like with the first isomorphism theorem of groups we often write this as r mod the kernel of a map psi this is isomorphic to the image of psi and so in a nutshell that is what the first isomorphism theorem tells us right here r mod i is isomorphic to the image oftentimes you attach to the first isomorphism theorem the following commutative diagram so you start off with this right here so you just have an arbitrary ring homomorphism between two arbitrary rings r and s well since this is a ring homomorphism psi right here it has a kernel and that kernel is going to be an ideal so from psi we can then construct this natural map where we mod out the kernel of psi aka the ideal i here that's what this natural map is doing and so the first isomorphism theorem then says that given this map psi and then we can add to it um aida right here you can always uniquely fill in the triangle here so you form what we call a commutative diagram commutative diagram means that no matter which path you take it's the same one so that you could take the direct path to r and s or you could take a detour through the quotient group r i r mod i v and then you can use that detour to get to s whichever path you take it's the same thing and so given any ring homomorphism here you have this unique homomorphism that'll go from r mod i to s that allows the diagram to be commutative the first isomorphism theorem is extremely important groups uh in group theory it's also important by analog and ring theory a lot of proofs that involve um quotient rings you utilize the first isomorphism theorem absolutely now the proof of the first isomorphism theorem for rings is essentially the same proof as it was for groups for which take a look at the take a look at the attached video if you want to see the proof of that again but all of the appropriate parts just change over from from rings to groups uh and from groups to rings it has no different of an argument it has to do of course um with just how these functions are constructed what does the natural map do and then how can you build this map fee it really comes down to since the quotient structure of the ring is well defined everything works out perfectly we also have in addition to the first isomorphism theorem for rings we also have the second and third isomorphism theorems as well for which what would that look like in the language of ring theory let r be a ring in which case we're just assuming this is a ring it's a associate of multiplication we're not assuming it's commutative or not assuming it's identity or anything like that although those axioms could be stacked on potentially um so let r be a ring with a subring i and an ideal j so this is an important thing here j is going to be an ideal here i is just an arbitrary subring so it's closed under multiplication it doesn't necessarily have the strong ideal closure per se uh but in that situation you're then going to have that the quotient ring i plus j mod j is isomorphic to i mod i intersect j like so and so i want you to try to compare this to what the second isomorphism theorem tells us in group theory uh because in group theory we talk about a subgroup and a normal subgroup they don't both have to be normal in that situation every ideal is a subring of the ring but not all subrings are necessarily ideals and so we can form a subring by taking the sum of two subrings so i plus j is a subring of r and that subring i plus j will contain the ideal j and it'll be an ideal for i plus j as well since it's an ideal of r it'll be ideal for any subring that contains it so this quotient ring does make sense and then you can compare that over here if you take a subring an ideal and you intersect them that'll form an ideal for that subring you have an analogous statement for the intersection of a subgroup with a normal subgroup in group theory and these two these two quotient rings are gonna be the same thing and i can say in this situation that the proof is literally the same because the proof is basically the following the proof is drum roll the first isomorphism theorem right that is the proof of the second isomorphism theorem now of course there's some mechanics you have to do you have to define of course a homomorphism you have to define a homomorphism between the the two the two rings or something so you might be like oh size can be a map from i over two i plus j over j there's of course other ways you could do it and then you you talk about this homomorphism you show that it's on to um because after all we're going to be isomorphic to the image then you have to compute the kernel and the kernel of size can equal j and then you invoke the first isomorphism isomorphism theorem so yeah there are some different ingredients between groups and rings but it's very immaterial the second isomorphism theorem comes as essentially immediate consequence of the first isomorphism theorem so if you have a theory of algebra that has the first isomorphism theorem then you're going to get the second isomorphism for three right and the same thing for the third isomorphism theorem remember what this one says if we have a ring r and i and j are both ideals of r in particular j is a sub ideal of i right there then we have that if you take the quotient rings r mod j if you mod that by our i mod j which i mod j naturally is an ideal of r mod j this is isomorphic to r mod i so in some essence you can simplify the fractions just get r mod i this the proof again comes from the first isomorphism theorem you come up with a map psi where you can go from r mod j over to r mod i you argue that this map is on too so that the image is all of r mod i then you compute the kernel and you show the kernel here is going to equal i mod j and then you invoke the first isomorphism theorem it's the exact same proof for rings just like it was for groups and this is very important principle here because this kind of shows us a first inkling of what's called universal algebra because as we study different algebraic categories in our lecture series if you include abstract algebra one and two we've talked mostly about groups and rings but amongst the groups we can be more specific we've talked about p groups and abelian groups and etc for rings of course we talk about rings and commutative rings and integral domains and fields and skew fields lots of different families in there and then there's generalizations i've often mentioned here or there right so we can generalize the notion of a group because we can talk about a semi group or a quasi group or a heap and these these generalizations and ultimate algebraic structures we'll talk about some other things later on this lecture series like lattices and boolean algebras and the like we have all these different theories of different algebraic structures vector spaces again just to name a few of these things we could talk about there are some things about different algebraic categories that are universal that is they work the same in every algebraic setting because there's something the same about everything and one of those first observations are the isomorphism theorems the first second and third isomorphism theorems hold for any algebraic category for which i'm not going to define what that means specifically but basically any theory of algebraic objects analogous to groups and rings and such you can get these three isomorphism theorems and it's the exact same proofs you use for group theory you just have to of course change the appropriate parts like what's what's a homomorphism mean in this setting what's a quotient algebra in this setting what's a kernel in that setting but if you can make sense of all of those then the proofs are all the same and of course it shouldn't it's we shouldn't go without also talking about the correspondence theorem right sometimes called the fourth isomorphism theorem we talked about this for groups what's the analog for for rings let i be an ideal for a ring r then there's a one-to-one correspondence between the set of sub rings of r containing i and the subset of ring sub rings excuse me the set of sub rings of r mod i and so there's this correspondence you take all of the sub rings of r that contain the ideal those will coincide with all the sub rings of r mod i and the relationship is you just send that sub ring of r that contains i to its quotient ring j mod i so in this situation j is just a sub ring this is a bijective map um furthermore this correspondence restricts to a one-to-one correspondence between the set of ideals containing i and the set of ideals of r mod i so we saw the same thing with groups that you can there's this correspondence between normal sub groups in which case you can lift up you can lift up a normal sub group of the quotient and that gives you a normal sub group of the original one the same is also true for rings if you find an ideal of r mod i that'll lift to an ideal of r that contains i and you can go from there there are many induction arguments we can make with rings because of something like this we can reduce a ring to its quotient maybe use use the induction hypothesis because it's not how smaller order we can lift it back up um there's of course non-induction arguments there as well but this idea of lifting lifting a property from the sub ring from the quotient ring i should say to the larger ring is a very important property which is why the correspondence theorem will be very useful for rings as well this idea of universal algebra is not only restricted to the isomorphism theorems but this is one of the first examples where it's like these are literally the same theorems we did for group theory but again the appropriate parts changed like normal sub groups are now ideals and things like that so this really is what you could sort of call meta abstract algebra we're starting to abstract what's already abstract algebra it's a pretty cool field of study and invite everyone watch this video just take a look into it sometime if you are curious