[Math] does simple ring implies artinian ring

noncommutative-algebra

so my doubt is that i am studying wedderburn artin theory and it gives structure of simple artinian rings, but if a ring is simple, it has no nonzero proper 2-sided ideals so it satisfies DCC on ideals trivially, so must be artinian, so if every simple ring is artinian, means every simple ring is of the form a matrix ring over a division ring, in T.Y Lam he studies simple left artinian first and then says it is same as simple right artinian and so we can talk about simple artinian straightaway only.

so is there a simple ring which is not artinian?

Best Answer

The Weyl algebra $A_1 := k\{x, y: xy - yx = 1\}$ is simple when char $k = 0$ but is not Artinian: the left ideals $A_1x^i$ form an infinite decreasing sequence of left ideals which is never constant.

Motivation for the Jacobson radical

Well, it sort of proves its own usefulness by being at the heart of so many algebraic theorems. But if you insist, there are a few good reasons that it is interesting.

For one thing, it is the largest ideal such that R/J has “the same simple right(/left) modules.” Looked at another way, it is the set of elements that don’t tell us anything about the simple modules (since they annihilate all simple modules.)

Motivation of the Jacobson density theorem

The Artin-Wedderburn theorem classifies which rings are like vector spaces in that they split into direct sums of simple modules. In particular, it says that simple Artinian rings are rings of linear transformations of finite dimensional vector spaces over division rings. What could be more natural? This type of ring is studied by undergraduates in linear algebra.

The Jacobson density theorem extends this result on simple Artinian rings to a larger class of rings called right(/left) primitive rings. A right primitive ring is a ring with a simple module with trivial annihilator. Rings of linear transformations of (right)-vector spaces over division rings (any dimension) are all right primitive, but this actually is a proper subset of all such rings. All of the rings that are “dense” in such rings are also right primitive.

The adjective “dense” is fitting because it is used in the topological sense. $R$ being dense means that (under a particular topology on the ring of linear transformations of a vector space over a division ring) that every nonempty open set contains an element of your ring $R$. In particular, given any linear transformation, there is a sequence of elements of your ring approximating that transformation.

Why do modules play an important role?

For you, approaching from the angle of representation theory, the most convincing thing might be that the F-representations of a group G are in one-to-one correspondence with the right $F[G]$ modules.

Really, there are lots of other reasons that modules are informative about their rings. That is basically the premise of homological algebra. You might check out Anderson and Fuller’s Rings and categories of modules

I feel the proof of some theorems are tricky and elaborate

Where is the rule that the theorems are simple and straightforward? What do you imagine they should be? Of course, this illusion of complexity in any field usually dissipates with growing exposure to the field.

If you really want to understand a particular theorem better, just find it in as many books as possible and compare the proofs and expositions. Usually this makes things come together faster. You should really check out Jacobson’s Structure of rings and Basic abstract algebra I+II since he was a pretty good expositor on structural theorems.

I can’t see the big picture of ring theory

Well, very few people can say that they understand the entirety of their field of mathematics. Most fields are just too large now. The classical structure theorems that you have already mentioned are nice examples (but hardly the pinnacle) of structural ring theory. If you want some nice books on general ring theory check out Lam’s First course in noncommutative rings or Faith’s Rings and things.

Should I read Hungerford again? Or just continue studying representation theory?

Whether or not you’re satisfied with the ring theory before you move on depends on your temperament and needs. Personally speaking I haven’t bothered referring to Hungerford since my qual exams. I never found Hersteins’ book useful, and I haven’t had the pleasure of coming in contact with McCoy’s book yet. On top of the books I’ve already mentioned, you might like Carl Faith’s or Louis Rowen’s volumes on ring theory as general references. If you check out the content on the Artin-Wedderburn theorem and the theory of right primitive rings in all the books I mentioned, I think you’d feel much better about the theorems, but spend your time as you see fit, of course.