This topics in algebra course will be a basic introduction to the theory of noncommutative
rings. The main reference for the core material covered in the first part of the course
will be the textbook ``An introduction to noncommutative noetherian rings" by Goodearl and Warfield.
We will cover parts of chapters 1-6 and 10 of the text, including skew polynomial rings; the prime spectrum; semisimple
rings and Wedderburn theory; and rings of fractions (including Goldie's theorem).
The second part of the course will be an introduction to the theory of finite dimensional division algebras, which
is not covered in the Goodearl text. A good reference for this part is the book "Noncommutative algebra" by Dennis and Farb.
(In the last week, we gave a few introductory lectures on Artin-Schelter regular algebras.)
The prerequisite for the course is Math 200 or Math 202, or an equivalent background. Concurrent enrollment in one of these courses is possible with permission of the instructor. The course is not recommended for a student with no graduate level algebra background. We will assume the basic ring theory and module theory that is part of Math 200 or math 202.
Homework sets will be posted to this website due roughly every two weeks. To get the most out of the course, it is important to review course notes and to do some homework problems.
**Important**
I have decided to clarify the grading policy for the course. The course grade will be determined based on attendance and performance on the homework. I have decided I would like homework to be handed in. A registered student is expected to attend with regularity and to make a significant effort on some portion of the homework problems (though completing all posted problems is not necessarily required.) A student who wants to learn about this material but does not feel they will be able to satisfy this homework requirement should audit instead of registering for this course. Please feel free to come talk to me if you have questions about this policy.
**Important**
Lecture Summaries
Lecture 1 (1/09/12): Introduction. Various important classes of noncommutative rings
with applications to other parts of mathematics: group algebras, rings of differential operators, universal enveloping algebras, finite-dimensional division algebras.
Lecture 2 (1/11/12): Introduction to skew polynomial rings (Ore extensions). Rings of the form R[x; alpha] for an endomorphism alpha and
the special example of the quantum plane. Differential operator rings of the form R[x; delta] for a derivation delta and the special
example of the first Weyl algebra. The definition of the most general skew-polynomial ring R[x; \alpha, delta] for an endomorphism alpha and
alpha-derivation delta.
Lecture 3 (1/13/12): Proof that R[x; \alpha, delta] exists. For a division ring D, D[x; \alpha, \delta] is a left PID and is a right
PID if alpha is an automorphism. Noetherian rings. If alpha is an automorphism, the skew Hilbert basis theorem says that if R is left (or right) noetherian, then R[x; \alpha, delta] has the same property (proof was omitted).
NO CLASS (MLK day) (1/16/12)
Lecture 4 (1/18/12):
The Weyl algebra A_n, defined as an iterated Ore extension. Proof that the Weyl algebra is a simple ring over a field of characteristic 0. More generally,
proved if R is a Q-algebra with an outer derivation delta such that R has no proper delta-ideals (R is delta-simple) then R[x; delta] is a simple ring.
Lecture 5 (1/20/12): The noetherian property. Bimodules. Formal triangular matrix rings. The use of this construction to find rings which are noetherian on on one side only. Prime ideals and their various characterizations.
Lecture 6 (1/23/12): Characterizations of semiprime ideals I: I is an intersection of prime ideals (the def.); J^n contained in I implies J is contained in I;
xRx contained in I implies x is in I. Prime ideals in the quantum plane k[x][y; sigma] where sigma takes x to qx for q not a root of 1. Comments on the case where
q is a root of 1.
Lecture 7 (1/25/12): The Jacobson radical and various characterizations as the intersection of all left max ideals or the intersection of all primitive ideals
(or either of the same right-sided intersections.)
Lecture 8 (1/27/12): Semisimple rings and the Wedderburn theorems. A ring is semisimple (all of its left module are semisimple modules, i.e. direct sums of
simple modules) if and only if it is a direct product of finitely many matrix rings over division rings. Semiprimitive artinian rings are semisimple. Semiprime artinian rings
are semisimple. A artinian ring is noetherian.
Lecture 9 (1/30/12): Localization of a ring R at a multiplicative system S of nonzerodivisors. Theorem: RS^{-1} exists if and only if S is a right Ore set,
that is, aR \cap bS is nonempty for all a in S, b in R. Sketch of naive proof of theorem (but will prove later using injective modules).
Lecture 10 (2/01/12): The special case where S is all nonzero elements in a domain R, then R is called a right Ore domain if S is a right Ore set. If a domain
does not contain a copy of the free algebra on two generators then it is an Ore domain. GK (Gelfand-Kirillov)-dimension. A ring of finite GK-dimension is an Ore domain.
Lecture 11 (2/03/12): Injective modules. Every module is contained in an injective module. Essential extensions. Injective hulls. Every module has an injective
hull which is essentially unique.
Lecture 12 (2/5/12): Proof that the localized ring RS^{-1} exists when S is a right Ore set, using injective hulls.
Lecture 13 (2/8/12): Goldie's theorem: A semiprime noetherian ring has a set of regular elements S which is a right Ore set. proof part (1).
It is enough to show that a right ideal is essential if and only if it contains a regular element. Goldie rank. Proof using Goldie rank that
a right ideal containing a regular element is essential.
Lecture 14 (2/10/12): Goldie's theorem proof part 2. An essential right ideal in a noetherian ring contains a regular element. In fact
Goldie's theorem works as long as the ring has finite rank and ACC on right annihilator ideals; these conditions together are called right Goldie. In fact these conditions are equivalent to the set S of regular elements being an Ore set (didn't prove.)
Lecture 15 (2/13/12): The Quaternions H. Division rings. Central simple algebras. A central simple k-algebra is
of the form M_n(D) where D is a central simple k-algebra, and n and D are uniquely determined.
Lecture 16 (2/15/12): R, S k algebras, then Z(R \otimes_k S) = Z(R) \otimes Z(S), and if R is simple and S is central simple
then R \otimes_k S is simple. In particular, a tensor product of central simple algebras is simple. A central simple k algebra has dimension over k
which is a perfect square. S central simple, then S \otimes S^{op}
is isomorphic to M_n(k) for n = [S:k].
Lecture 17 (2/17/12): Central simple algebras over k are called similar if they have the form M_m(D) and M_n(D) for the same central division algebra D. The Brauer group Br(k) is the set of similarity classes of central simple k-algebas. Proof this is a group under tensor. Some examples of Brauer groups without proof. Generalized quaternion algebras (a,b/F) for a field F. Criterion for when this is a division algebra and when it is split (isomorphic to M_2(F)).
NO CLASS (Presidents' Day) (2/20/12):
Lecture 18 (2/22/12): Skolem-Noether theorem. Application: All central simple algebras of degree 2 (i.e. of dimension 4 over their centers)
are generalized quaternion algebras (in characteristic not 2).
Lecture 19 (2/24/12): Double centralizer theorem. Application: every maximal subfield L of a division algebra D with center k has [L:k] = deg(D).
Classification of division algebras over the real numbers (Frobenius's theorem).
Lecture 20 (2/27/12): If K is a strictly maximal subfield of S, then K splits S. Jacobson-Noether theorem. Every central simple algebra S is split by some Galois extension of k (which we cannot necessarily take inside S itself, even if S is a division algebra.) If K splits S then K is a strictly maximal subfield
of some unique central simple algebra T equal to S in the Brauer group. The relative Brauer group. Br(k) = union Br(K/k), the union over all finite Galois extensions of k.
Lecture 21 (2/29/12): Crossed product algebras. Every central simple algebra with a strictly maximal subfield K galois over the base field k is a
crossed product algebra, and conversely.
NO CLASS (Prof. Rogalski out of town) (3/2/12, 3/5/12):
Lecture 22 (3/7/12): Review of crossed product algebras. There is a bijection between Br(K/k) and equivalence classes of factor sets. The special case of
cyclic algebras. The cylic algebra (K, G, a) is isomorphic to K[x; \sigma]/(x^n -a). Quaternion algebras are cylic algebras and calculation of the factor set in this case.
Lecture 23 (3/9/12): Ext. Cohomology of groups. Examples when G = Z or G is cyclic. The bar resolution.
Lecture 24 (3/12/12): The relative Brauer group Br(K/k) is isomorphic to the cohomology group H^2(G; K^*). Group cohomology
is torsion, so the Brauer group is torsion. A worked example: Br(Q(i)/Q).
Lecture 25 (3/14/12): Introduction to Artin-Schelter regular algebras. Graded rings and modules. Minimal projective resolutions and Global dimension.
Lecture 26 (3/16/12): Classification or AS-regular algebras of global dimension 2. Brief discussion of the ideas behind the classification of AS-regular
algebras of global dimension 3.