# Spring 2023

### Time and Place

• Lectures: MWF 12:05pm – 12:55pm Carver 132

### Examination

Each week a homework set is due, except first week and the week of an exam. There will be two non-cumulative take-home exams and no final.

Dates for the exams are to be determined.

### Homework Sets

1. (due Fri 1/27) 1. Let $$f:R\to S$$ be a ring homomorphism. (a) Prove that if $$M$$ is an $$S$$–module then $$M$$ becomes an $$R$$–module by defining $$r.m=f(r).m$$. (b) Prove that if $$\varphi:M\to N$$ is an $$S$$‐module homomorphism, then $$\varphi$$ is automatically an $$R$$–module homomorphsm, when $$M$$ and $$N$$ are turned into $$R$$–modules as in part (a). §10.1: 9, 19, 21.
2. (due Fri 2/3) §10.2: 9, 13 (Hard. First assume $$I^2=0$$); §10.3: 7, 9; §10.4: 6.
3. (due Fri 2/10) §10.4: 11, 16, 22, 24, 25.

### Lecture Summary

#### Part I: Modules

Chapters 9–12

In the first part of the course we study modules. These are generalizations of abelian groups and vector spaces and play an important role in representation theory, differential geometry, and other areas. Beyond the basics, key goals include:

• tensor products and multilinear algebra
• finitely generated modules over PIDs (with applications to normal forms of matrices)
• some category theory to help us see the unity between constructions

Below is a short summary of each lecture.

1. Definition of modules. Examples.
2. More examples. Algebras. Group representations. Submodules. Direct sums (coproducts) and products.
3. Universal property of products and coproducts. Free modules.
4. Quotient modules and isomorphism theorems.
5. Presentations of modules. Restriction and induction (examples/motivation).
6. Free modules, definition of induced modules $$S \otimes_R M$$.
7. Tensor products of general bimodules. Balanced maps and the universal property of tensor products.
8. Examples: Computing $$\mathbb{Z}_n\otimes_\mathbb{Z}\mathbb{Z}_m$$. The "tensor unit" isomorphism $$R\otimes_R M\cong M$$.

#### Part II: Fields

Chapters 13–14

In the second part of the course we study fields. Since every homomorphism between fields is injective, one is led to study subfields and field extensions. Galois theory provides a link from this to group theory. Historically, Galois theory led to the resolution of several problems in Euclidean geometry that had been open for thousands of years.

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