Wrapping up the Jacobson Radical

We now have the following equivalent definitions of the Jacobson radical. Remember right now we assume R is commutative with 1.

1) Intersection of all maximal ideals
2) Intersection of the annihilators of all simple left R-modules
3) The set of non-generators of R
4) The set of elements, x, such that 1-rx has a left inverse for all r.

I think I already pointed out that from at least two of these definitions we automatically get that J(R) is a two-sided ideal. Two basic examples are now that if R is any field, then J(R)=\{0\}. And if K is a field, and R=K[[x_1, \ldots x_n]], then J(R)=\{f\in R : f \ has \ 0 \ constant \ term\}. An important generalization is that in any local ring the Jacobson radical is the set of non-units.

An important result called Nakayama’s Lemma is that if M is finitely generated, then M=\Phi(M)+N implies that M=N. Special case: If M= J(R)M+N, then M=N. Corollary to that special case: If M=J(R)M, then $M=\{0\}$ (this last form is what is sometimes called Nakayama’s Lemma).

Proof: Suppose M=\langle x_1+n_1, x_2+n_2, \ldots, x_m+n_m\rangle. Where x_j\in \Phi (M), and n_j\in N for all j. Define S=\{n_1, \ldots, n_m\}.

Then with this setup, we exploit the non-generator definition. Note that
M=\langle x_1, n_1, x_2, n_2, \ldots, x_m, n_m\rangle
= \langle S, x_1, \ldots, x_m\rangle
= \langle S, x_1, \ldots, x_{m-1}\rangle
… etc
= \langle S\rangle \subset N.

And we are done! It may have seemed a little roundabout to go through the “Frattini submodule” in developing the Jacobson radical, but it certainly pays off to have lots of definitions as we see here.

The last little bit I wanted to say was that we can define the Jacobson radical for a ring without identity. I don’t want to go through the details, but a standard trick is to define a new ring (with identity) S=\mathbb{Z}\times R with the standard addition, and then (a,b)(c,d)=(ac, ad+cb+bd). It is pretty basic to check that J(S)=\{0\}\times I where I is some ideal in R (by the fact that J(\mathbb{Z})=\{0\}). It is also just algebraic manipulation to check that I is the largest ideal in R such that for every x\in I there is a y\in I such that x+y-yx=0. This then is our definition. J(R)=\cap_{\mathfrak{I}} I where \mathfrak{I} is the collection of ideals satisfying that property.


The Jacobson Radical Part II

First recall that we showed J(R)=\Phi (R), and hence is a submodule of R as a module over itself. Thus J(R) is a left ideal of R. Next recall that we showed J(R)=J(R)R, and hence is a right ideal. i.e. J(R) is a two-sided ideal.

Let’s now work towards the annihilator definition. Define an equivalence relation the set of maximal ideals of R by I ~J if there is a simple left R-module M with elements a,b\in M such that I=ann_R(a) and J=ann_R(b). We see that this is an equivalence relation, since I ~ J iff R/I and R/J are isomorphic as R-modules. Examine the module homomorphisms r\mapsto ra and r\mapsto rb to see that if I ~ J then R/I\cong M \cong R/J. Also, if R/I\cong R/J by the iso \varphi, then J=ann_R(\varphi^{-1}(1+J)), so I ~ J since \phi^{-1}(1+J), 1+J\in M with I=ann_R(1+J) and J=ann_R(\varphi^{-1}(1+J)).

Now let \mathfrak{I} be an equivalence class of maximal left ideals. I claim that \cap_{I\in\mathfrak{I}} I=ann_R(M), where M is a simple left R-module isomorphic to each R/I, for I\in\mathfrak{I}. By definition and the property above we get that \mathfrak{I}=\{ann_R(a): a\in M, \ a\neq 0\}. Thus if J\in\mathfrak{I}, then J=ann_R(1+J) which means that J=ann_R(a) where \varphi: R/J\to M satisfies \varphi(1+J)=a. But now this gives precisely cap_{I\in \mathfrak{I}}I=ann(M).

Now just intersect over all the maximal left ideals. We get \displaystyle J(R)=\cap_{J \ maximal} J=\cap_{\mathfrak{I}}\cap_{i\in \mathfrak{I}} I=\cap_{\mathfrak{I}}ann_R(R/I)=\cap_{M \ simple}ann_R(M). And voila, we have it. This was a rather terse run-through and assumed a working knowledge of some facts about modules, but I find it to be a rather fascinating take on the development.

Next we’ll exploit some of these definitions to get some properties of the Jacobson radical, and develop it in a method that doesn’t require our ring to have 1.

An Approach to the Jacobson Radical

I’ve decided that in order to better understand the concepts in algebra this quarter, I should probably start posting several times a week on them. The quarter system is fast, and so we only have three weeks left.

I want to develop the Jacobson radical (this is really the next step in all that stuff I was presenting before anyway). So at first we’ll assume our ring R has 1 (which was the case before). The typical development seems to define \displaystyle J(R)=\bigcap_{M \ simple \  R-module}ann_R(M). The Jacobson radical is the intersection of the annihilators of all simple left R-modules (note that the ann_R(M)=\{r\in R : rm=0 \ \forall m\in M\}.

There are many, many alternative formulations, but I want to develop this from one of the more obscure angles.

First, let \Phi(M) be the “Frattini submodule” i.e. the set of nongenerators. x is a nongenerator if we have any subset of M, say S, and R\langle S, x\rangle= M, then R \langle S\rangle = M. So a nongenerator means that if you ever have a set which generates your module, then that set without the nongenerator will still generate it.

Step 1: \Phi(M)=\bigcap_{N<M \ maximal}N.

Suppose x\notin \Phi(M). Claim: there is a maximal submodule N such that x\notin N. Let S be such that M=R\langle S, x \rangle and M\neq R\langle S\rangle. Then since x is not a nongenerator we can choose S so that x\notin R\langle S\rangle. Now R\langle S\rangle \in P=\{H : H<M, S\subset H, x\notin H\}. We have a non-empty partially ordered set, and unioning gives an upper bound to any chain. Thus we apply Zorn’s Lemma to get a maximal element, N. Thus \displaystyle \Phi(M)\supset\bigcap_{N<M \ maximal}N.

For the reverse, just note that if x is not a member of the right side, then there is a particular maximal submodule, N, such that x\notin N. Thus M=R\langle N, x\rangle, but R\langle N\rangle=N. So x\notin \Phi(M). And we get equality \Phi(M)=\bigcap_{N<M \ maximal}N. So the Frattini submodule equals the intersection of all maximal left R-modules.

Step 2: Define the Jacobson radical now to be J(R)=\{x\in R: 1-rx \ has \ left \ inverse \ \forall r\in R\}. Claim: \Phi(M)\subset J(R).
Suppose x\notin J(R). Then there is some r\in R such that 1-rx has no left inverse. i.e. 1\notin R(1-rx)\subset N where N is a maximal left ideal. But 1-rx\in N since 1\in R. Thus x\not in N because otherwise would mean that 1=(1-rx)+rx\in N a contradiction. So x\notin \Phi(R). Note the subtle shift in usage here. We are considering R to be an R-module over itself.

Step 3: J(R)M\subset \Phi(M).

Suppose x\in J(R) and m\in M. Claim: xm\in \Phi(M). Suppose that M=R\langle S, xm\rangle and m\in M. Then we have that m=\sum (r_js_j) +rxm. i.e. m-rxm=\sum r_js_j\Rightarrow (1-rx)m=\sum r_js_j.

Suppose b(1-rx)=1 (since x\in J(R)). Now m=b(1-rx)m=\sum br_js_j\in R\langle S\rangle. Thus xm\in R\langle S\rangle\Rightarrow R\langle S\rangle=R\langle S, xm\rangle = M. And we are done.

Now notice from the above steps that \Phi(R)\subset J(R)=J(R)\cdot \{1\}\subset J(R)\cdot R\subset \Phi(R). And so we have equality all the way through. Namely, \Phi(R)=J(R). So the set of all nongenerators, the intersection of all maximal submodules, and the set of elements such that 1-rx has a left inverse for all r\in R are all the same (namely, the Jacobson radical).

Next time we will bank on these to show that these are all equivalent to the first listed and more common definition, the intersection the annihilators of all simple left R-modules.

(I claim no responsibility for the invention of this approach. This is the way my professor sees the world).

Lost in the Funhouse

I’m in the position where I’m too tired to start thinking about a new algebra problem, but it is too early to legitimately call it a night, so I’ll just shift gears and do a new post. It has been one of those days where about everything that can go wrong does. So how about thinking about a little humor.

A very important work of literature in the “post-modern canon” is John Barth’s Lost in the Funhouse. I was first led to it by the last story in DFW’s collection Girl With Curious Hair. He does a sort of parody/hommage to it. Anyone that has taken a 20th century American lit course has probably had to read something by Barth, and it was most likely the title story in this collection.

Barth is known for his excessive meta-fictional devices and influence on writers mentioned previously like Pynchon, Wallace, and probably any serious post-modernist. Despite the term “excessive meta-fictional devices,” I find him quite easy and fun to read. The devices serve a purpose and are usually humorous. Unlike some post-modernists that came after him, Barth is very much concerned with art expressing a human experience (mostly love). Although DFW ultimately rejected Barth, he very much agrees on this point…but we’ll get to that later.

I think I’ll mostly do this as an analysis of the title story. Lost in the Funhouse is a short story “about” a boy’s (Ambrose) trip to Ocean City where he enters a funhouse, and yes gets lost. He enters with a girl Magda, but she continues on with his brother while he is left alone. But really it is not about this at all. The funhouse serves as a metaphor for Ambrose’s first sexual experience. In my reading, I actually don’t believe the trip happens at all, and the whole entire trip is a metaphor.

In the typical Barth fashion, the funhouse is a multi-layered metaphor. A funhouse has mirrors all around. This means that Ambrose must see himself reflected in all shapes and sizes. This represents his fractured subconscious about the experience. His own head is also in the way of ever directly seeing the image in the mirror behind him. This aspect of the metaphor is actually extensively rejected by Wallace. Wallace interprets that aspect as Barth’s way of saying that literature can never directly make it to the reader. It always will hit the reader’s head first and be obscured and never directly viewed. “…that the necessity for an observer makes perfect observation impossible, …” Wallace changes the metaphor and says it is like a bow and arrow. Your arm will always be in the way of shooting directly, but the writer can take this into account and directly hit the reader.

This is all boring, though. Let’s get to the truly interesting aspects of the story. Barth as a narrator sometimes narrates, sometimes talks directly to the reader, and sometimes comments on the narration. It is these comments that are the humorous meta-fictional devices. The story becomes self-aware. It understands and points out the devices it is using. Here is one of my favorite devices:

En route to Ocean City he sat in the back seat of the family car with his brother Peter, age fifteen, and Magda G____, age fourteen, a pretty girl and exquisite young lady, who lived not far from them on B_____ Street in the town of D____, Maryland. Initials, blanks, or both were often substituted for proper names in nineteenth-century fiction to enhance the illusion of reality. It is as if the author felt it necessary to delete the names for reasons of tact or legal liability. Interestingly, as with other aspects of realism it is an illusion that is being enhanced, by purely artificial means.

The story is continually interrupted to go off on tangents like this. He wants to point out, explain, and make fun of the traditional devices he is using. In doing this he is actually creating new and original devices. He doesn’t want the reader to become absorbed in the story and think that it is real for the duration. He wants the reader to be painfully aware that they are reading a story.

Another aspect of the verbal trickery of the story is to somehow assert the primacy of language to experience. All experience must be filtered through language. Thus, instead of ever explicitly describing Ambrose’s experience, we only live on the verbal thoughts flowing through his head throughout the experience. In fact, in searching for a certain quote just now, I came across another that reinforces my reading that the entire story is a metaphor.

With incredible nerve and to everyone’s surprise he invited Magda, quietly and politely, to go through the funhouse with him. ‘I warn you, I’ve never been through it before,’ he added, laughing easily, ‘but I reckon we can manage somehow. The important thing to remember, after all, is that it’s meant to be a funhouse; that is, a place of amusement. If people really got lost or injured or too badly frightened in it, the owner’d go out of business.

Or even the famous opening lines, “For whom is the funhouse fun? Perhaps for lovers. For Ambrose it is a place of fear and confusion.” Don’t read that as “funhouse.” Ambrose is really talking about the fact that it is his first sexual experience. He is trying to convince Magda that it can’t be too scary painful, since people continue to have sex. The funhouse is for lovers? It is scary and confusing for Ambrose? Come on, of course this is what it is talking about.

Alright. Let’s get back from that tangent to Ambrose’s head. He starts telling all of these scenarios of how his being lost gets played out. In one he actually dies. “This can’t go on much longer; it can go on forever. He died telling stories to himself in the dark; years later, when that vast unsuspected area of the funhouse came to light, the first expedition found his skeleton in one of its labyrinthine corridors and mistook it for part of the entertainment. He died of starvation telling himself stories in the dark;…” This all emphasizes the main effect Barth is striving for. All human experience is mediated by language. Language is so primary and important that a mind preoccupied by other stories could completely miss the experience itself.

It seems I’ve gone on longer than I should have, but I feel like I haven’t done the story the slightest bit of justice. It is so great and packed full of interesting things. And this is just one of many stories in the book. I highly recommend this to anyone who aspires to understand modern literature.

Andrew Bird/Duncan Sheik Joint Post

This amazing and awful thing happened this year. The two people that I feel a strong need to buy an album of on the day it releases both released their new albums at the end of January. These two people are Duncan Sheik and Andrew Bird.

Let’s start with the Bird. I am firmly convinced that this is his best album yet. This is a bit of a relief, since I felt that his previous one took a step down. I thought that his previous one fell massively in lyric creativity, and was much more mainstream musically. Although I was very happy in the direction he was heading with overall sound. That’s why I always recommend Mysterious Production of Eggs for someone interested in getting to know his music.

This album did the perfect combination. It kept going in the interesting sound that I liked from the last one, but brought back the lyrical inventiveness and quirkiness of his earlier stuff. In fact, I might argue that this album places Bird as the nation’s greatest creator of word play and association. Even the New York school of poets didn’t invent stuff this great. Here is something from “Anonanimal”:

I see a sea anemone
The enemy
See a sea anemone
And that’ll be the end of me.

Underneath the stalactites
The troglobites lost their sight

Sometimes the associations are much deeper than just in sound, though. I tend to classify Bird associations in the categories: juxtaposing two words that sound similar and have different meaning, juxtaposing two words that have similar meaning but don’t sound the same, or the most subtle of all, juxtaposing one word that conjures the other through societal associations. It is utterly amazing to hear this constant flow of associations that sometimes linger on an idea or flow through ideas that are completely unrelated except through sound. I tend to walk around as happy as can be.

Duncan Sheik brings a different side to things. If Bird is uplifting, nothing can be a bigger downer than this new Sheik album. Let’s just give two examples:

Lily keeps the lighthouse
She’s afraid of the unknown
She’s no ray of sunshine
So mostly she’s alone

No one cares about her longing
Or the dreams on which she’s fed
And though I’m sad to say it
She’d be better off dead

etc…as it runs through many scenarios in which people would be better off dead. The harshness doesn’t end there, though:

Life is like a sinking ship
And you are at the wheel
See the hulls filling up
You know your fate is sealed

Still you keep on trying
To steer the ship to shore
It’s time to learn a lesson
You should’ve learned before

There’s nothing you can do
And we don’t believe in you

I must admit that at first I was so engulfed by this album that I couldn’t even bring myself to listen the Bird for the first time. I was in some sort of shock that someone could write music like this. It moved and disturbed me. In the long run this album may end up getting a thumbs down, though. It is technically not a “Duncan Sheik” album, but him singing songs he wrote for some sort of musical or something.

I do like the on going narrative of the album, but the lyrics and music get a bit too music theater cliché at points. The album before this one was the cast recording of his musical Spring Awakening. The one before that wasn’t all that impressive to me either. That being said, it is without a doubt worth spending time with. I just have a feeling that it will get old fast and won’t be extremely friendly to re-listens. For the person unfamiliar with Sheik I’d start with either Humming (the most original and creative of his works in my opinion) or Daylight (the most polished and probably most listener friendly). Proceed from there as you wish.