MATH3800 (Fall 2025): Algebra.

Instructor: Prof. Ivan Loseu (email: ivan.loseu@yale.edu)

Lectures: MW 9-10.15am, location: KT 205. The first class is on 8/27, and the last class is on 12/3.

Office hours: in KT 715, M 10.30-11.30, 3.30-4.30, T 9.30-10.30.

This document contains some important information on the class content, homeworks, etc. Please read it!

Goals: This course serves as an introduction to Commutative algebra with a discussion of category theory and a brief discussion of connections to Algebraic geometry and Number theory. This document also contains a list of references. There is no required textbook.

Index of terminology.

Grading: Based on 5 homework sets (70%) and a 3.5 hours final exam (30%). Due dates for homeworks are Sept 16, Oct 7, Oct 28, Nov 16, Dec 4. The final will be on Dec 16 at 2pm.

  • Homework 1, due Sept 16.
  • Homework 2, due Oct 7.
  • Homework 3, due Oct 28.
  • Homework 4, due Nov 16.

    Rules: For homework sets, peer discussion is allowed but only after your own thinking; you are also required to write your solution on your own, if it (partially) comes from a discussion. The only allowed source is lecture notes, either your own or the notes by the instructor that are going to be uploaded on this webpage. Homeworks sets are to be submitted via Canvas as a pdf upload. You are encouraged to texify your work, but scans of handwriting are also acceptable assuming it is clearly written. One unexcused late (by no more than a week) submission can be accepted, please email the instructor and provide an explanation. For subsequent late submissions, an official notice from a residential dean (for undergraduates) is required, otherwise the submssion will not be accepted.

    Preliminary list of topics together with their descriptions is in this document as well. You can also find references there.

    Schedule: The first chapter in this class is Basics of Commutative algebra.

  • Aug 27, Lecture 1: Rings, ideals and modules, part I, notes: The goal of this part is to introduce key concepts of this class. We will also be mentioning what's going to be studied later in the class. In this particular lecture, we introduce rings together with easy examples and constructions, ring homomorphisms, ideals and, time permitting, quotient rings.
  • Aug 29, Lecture 2: Rings, ideals and modules, part II, notes: we finish our discussion of quotient rings. We'll then discuss operations with ideals. The main part is a discussion about maximal ideals. We will also start talking about prime ideals.
  • Sept 3, Lecture 3: Rings, ideals and modules, part III, notes: We finish our discussion of prime ideals and start discussion of modules and their homomorphisms including examples and constructions.
  • Sept 8, Lecture 4: Rings, ideals and modules, part IV, notes: We continue our study of modules and their homomorphisms: we discuss the Hom modules, sub- and quotient modules, and also finitely generated modules, and free modules.
  • Sept 10, Lecture 5: Noetherian rings and modules, part I, notes: We discuss motivations to care about modules, and then talk about an important special class of rings (and modules) Noetherian one. We also state and prove one of foundational theorems in Commutative algebra, the Hilbert basis theorem.
  • Sept 15, Lecture 6: Noetherian rings and modules, part II/ Modules over PID's part I, notes: We finish our discussion of Noetherian modules, and introduce another class of modules/ rings -- Aritinian ones. Then we start discussing PID's (=principal ideal domains) and state the theorem classifying modules over them.
  • Sept 17, Lecture 7: Modules over PID's part II, notes: We prove the theorem from Section 3.3 of Lecture 5.
  • Sept 22, Lecture 8: Localization of rings and modules, I, notes: We prove the last proposition from Lecture 7 and start our discussion of localization of rings.
  • Sept 24, Lecture 9: Localization of rings and modules, II, notes: we finish the discussion of localizations of rings and discuss the localization of modules.
  • Sept 29, Lecture 10: Localization of rings and modules, III/ Integral and finite algebras, I, notes: We finish our discussion of localization of modules: we discuss interaction of localization with kernels and images and also with direct sums. We then start discussion of integral and finite algebras: we give basic definitions and examples and start proving basic properties.
  • Oct 1, Lecture 11: Integral and finite algebras, II notes: We finish the proof of the theorem from lecture 10 and then discuss the integral closures including the rings of algebraic integers.
  • Oct 6, Lecture 12: Integral and finite algebras, III, notes: We introduce an important class of domains, Dedekind domains, and prove that the rings of algebraic integers are such. The main nice property of Dedekind domains is that there is the unique factorization on the level of ideals. We'll state this theorem and, time permitting, discuss some ideas of proof.
  • Oct 8, Lecture 13: Connections to Number theory, notes: We prove the unique factorization of ideals in Dedekind domains and very briefly talk about class groups. Then we switch the direction -- we state and prove the Noether normalization lemma, that will be important in our discussion of connections with Algebraic geometry to follow in Lectures 14 and 15.
  • Oct 13, Lecture 14: Connections to Algebraic geometry, I, notes: We state and prove Hilbert's Nullstellensatz, define algebraic subsets in affine spaces and prove that they are in bijection with the radical ideals in the algebra of polynomials.
  • Oct 20, Lecture 15: Connections to Algebraic geometry, II, notes: We discuss irreducible algebraic subsets and show that they correspond to prime ideals. Then we discuss the decomposition into irreducible components. Then we define the algebra of polynomial functions on an algebraic subset. Finally, we discuss the importance of localization in Algebraic geometry.
  • Oct 22, Lecture 16: Categories, functors and functor morphisms, I, notes: We finish our discussion of the geometric significance of localization and proceed to the 2nd major topic for this course -- Category theory. We start it by a discussion of categories.
  • Oct 27, Lecture 17: Categories, functors and functor morphisms, II, notes: We define the other two basic notions: functors and functor morphisms and give many examples.
  • Oct 29, Lecture 18: Categories, functors and functor morphisms, III, notes: We state and prove the Yoneda lemma and discuss consequences, including the notion of objects representing functors. Then we apply this to products in categories.
  • Oct 31: Evil CATegorical pumpkin!
  • Nov 3, Lecture 19: Categories, functors and functor morphisms, IV, , notes: We discuss coproducts in categories. Then we define adjoint functors, give examples and prove a uniqueness result.
  • Nov 5, Lecture 20: Tensor products, I: We introduce the tensor product of two modules as an object representing the functor of bilinear maps and then construct the tensor product.