Math 510B Spring 2026
Instructor: C. Negron
Time and location: 12:00-12:50 pm, KAP138
Office: KAP 444C
Office hours: Fridays
Books: Part I) Finite dimensional algebras, Drozd and Kirichenko.
Part II) Introduction to commutative algebra, Atiyah and Macdonald and/or Stacks project Ch 10.
Part III) Back to Finite dimensional algebras, Drozd and Kirichenko.
Syllabus here.
lecture set 1, set 2
-- HWs, midterms, etc.
Quasi-weekly HWs, four or so quizzes, one midterm, one final.
-- How to do HW :)
Step 1: Read the problems.
Step 2: Try to do the problems.
Step 3: Succeed in doing some of the problems, and write them down, but fail to do others.
Step 4: Meet with your friends to share your collective insights.
Step 5: Ask the professor if anything remains unclear.
Step 6: Complete your homework.
Never: Search for solutions online/through an LLM.
-- Topics schedule
Part I) Algebras and modules → chain conditions → Jordan-Holder theorem → semisimplicity → Jacobson radical → Artin-Wedderburn → Maschke theorem → Artinian implies Noetherian
Part II) Noetherian rings → Polynomial rings and finite type algebras → Spectra and Zariski topology → nullstellensatz → localization and primes → nilradicals → finite extensions
Part III) Bimodules → tensoring and Hom functors → ⊗-Hom adjunction → Morita theorem → Projectives, injectives, and flat modules → constructing injectives
-- Homework
Report (presumed) errors via email or ask in class. To be turned in in class. Don't turn in garbage.
When writing up the HW, summarize the question as an introduction to each problem.
HW 1 (Due Fri Jan 23)
1. Let k be a commutative ring and A be a k-algebra. For any k-module M, prove that a choice of A-module structure on M is equivalent to the choice of a k-algebra map φ: A → Endk(M).
2. For any field k, prove that the only ideals in Mn(k) are 0 and Mn(k) itself. [Hint: Consider row and column reduction.]
3. Let k be a field.
(a) Prove that the standard module 𝕍 = kn over Mn(k) is simple.
(b) Prove that EndMn(k)(𝕍) = k·id𝕍. [Hint: Think about eigenvectors for the actions of the diagonals.]
4. Classify all simple modules over Q[x].
5. For any A-module M define the annihilator AnnA(M) ⊆ A as the subset consisting of all a in A for which a·M=0. Show that AnnA(M) is an ideal in A.
6. For any ideal I in an algebra A, with corresponding quotient map π: A → A/I, prove that the restriction functor
resπ: A/I-mod → A-mod
is fully faithful. Prove, furthermore, that this functor is an equivalence onto the full subcategory of A-modules which are annihilated by I.
7. (a) For any ideal I ⊆ A and A-module M, prove that the span I·M of all elements of the form x·m with x in I and m in M is a submodule in M.
(b) Prove that the quotient map πM: M → M/I·M induces an isomorphism of A/I-modules (A/I)⊗AM ≅ M/I·M. [Hint: Prove that πM has the requisite universal property.]