Groups, Rings, and Vector Spaces I: Topics list

Topics & Pacing

The following is a rough list of arguments covered in class. Suggested homework problems are listed within each entry.

Nov 21

Splitting short exact sequences. Homology. Snake lemma.
Homework: III.7: #1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 14, 15.

Nov 18

Midterm 2

Nov 16

Exact complexes. Short exact sequences.

Nov 14

Complexes of modules. Example: singular complex.

Nov 11

Noetherian modules.
Homework: III.6: #13, 14, 16.

Nov 9

Free modules and free algebras.
Homework: III.6: #5, 6, 7, 8, 9, 10, 11.

Nov 7

• Products and coproducts, kernels and cokernels in the category R-Mod.
  Homework: III.6: #2, 3.
• Remarks on Bimodules. Second Isomorphism Theorem for modules.
  Homework: III.5: #15, 16, 17, 18.

Nov 4

Canonical decomposition of a module homomorphism. First and third isomorphism theorems.
Homework: III.5: #12, 13.

Nov 2

Modules. Examples. R-algebras. The category of R-modules. Kernels, Submodules, Quotients.
Homework: III.5: #1, 4, 5, 6, 7, 9, 10, 11.

Oct 31

Prime and maximal ideals. Spec of a ring. Krull dimension.
Homework: III.4: #13, 14, 15, 18, 19, 20.

Oct 28

Products of ideals. Quotients of polynomial rings. Prime and maximal ideals.
Homework: III.4: #5, 6, 7, 10, 12

Oct 26

Nilradical. Sum, Intersection of Ideals. Noetherian rings. PIDs.
Homework: III.4: #2, 3, 11.

Oct 24

Characteristic of a ring. First and Third Isomorphism Theorems.
Homework: III.3: #10, 11, 13, 14, 15.

Oct 21

• Ideals and quotient rings.
  Homework: III.3: #2, 7, 8.
• Injective map from a ring to the ring of endomorphisms of the underlying abelian group (end of sect. 2).

Oct 19

Ring monomorphism equivalent to injective homomorphism (proof). Products and coproducts.
Homework: III.2: #12, 13, 14.

Oct 17

Universal property of polynomial rings.

Oct 14

• Ring homomorphisms. Monomorphisms equivalent to injectives (no proof, yet). Epimorphisms may not be surjections.
  Homework: III.2: #2, 4, 5, 8, 9.
• More on polynomial rings. Monoid and Group rings.
  Homework: III.1: #15, 16, 17.

Oct 12

Units (left, right, two-sided). Division rings. Quaternions. Fields. Introduction to polynomials rings.
Homework: III.1 #7, 10, 11, 12, 14.

Oct 10

Definition of ring. Zero divisors. Integral Domains.

Oct 7

Midterm 1

Oct 5

• Group objects in categories.
  Homework: II.10: #3, 4, 5.
• Every transitive action is isomorphic to the left multiplication on left-cosets. Lagrange's theorem for orbits.
  Homework: Same as previous class meeting.

Oct 3

• Group actions. Effective, free, transitive actions. Orbits.
  Homework: II.9: #2, 3, 4, 5, 8, 9, 11, 15, 17, 18.
• Cokernels, surjections, and epimorphisms in the category Grp.

Sep 30

Second Isomorphism Theorem. Lagrange's Theorem and examples of applications.

Sep 28

Aut(G), Inn(G), and Out(G).
First, Third, and Second (almost) Isomorphism Theorems.
Homework: II.8: #8, 11, 12.

Sep 26

Universal property of quotients. Commutator Subgroup. Presentations. Correspondence between subgroups of the quotient and subgroups containing a normal subgroup.
Homework: II.8: #4, 7, 10.

Sep 23

• Canonical decomposition.
  Homework: II.8: #1, 2.
• Equivalence relations, quotients by normal subgroups. Kernel ~ Normal.
  Homework: II.7: #10, 11, 12.

Sep 21

• Kernels as normal subgroups. Group structures and equivalence relations.
  Homework: II.7: #1, 5, 7.
• Subgroups of cyclic groups.
  Homework: II.6: #9, 12, 14.

Sep 19

• Subgroups, Kernels, Monomorphisms.
  Homework: II.6: #1, 5, 7, 15, 16.
• Free groups (reprise), Cayley graphs. Free abelian groups.
  Homework: II.5: #8, 9.

Sep 16

Free groups, universal property, construction of F(S).
Homework: II.5: #3, 6, 7.

Sep 14

Examples of group homomorphisms; isomorphisms; cyclic groups; homomorphisms of abelian groups.
Homework: II.4: #3, 7, 9, 10, 11.

Sep 12

Group homomorphisms; the category Grp.
Products and coproducts in Grp.
Homework: II.3: #6, 7, 8.

Sep 9

Examples of groups: permutation, dihedral, cyclic groups.
Homework: II.2: #1, 2, 3, 5, 6, 16, 17.

Sep 7

Basic properties of groups. Finite groups. Order.
Homework: II.1: #4, 9, 10, 11.

Sep 2

Morphisms, monomorphisms, epimorphisms, isomorphisms.
Universal properties. Products, Coproducts.
Homework: I.4: #3; I.5: #4, 6, 8, 9.

Aug 31

Definition of Category. Examples.
Homework: I.3: #1, 4, 6, 10

Aug 29

Reminders on naive set theory. Functions: injective, surjective, bijective functions.
Homework: I.2 #4, 6, 7.