(Topics for qualifying exam in algebra) 731 SYLLABUS
I. Setup (over rings with unity, including noncommutative)
Modules and Homomorphism Theorems
Direct Sums and Products, Free Modules
(including Universal Mapping Properties)
Projective and Injective Modules, Divisible Modules
Exact Sequences, Split Exact Sequences, Diagram Chasing
Hom and Tensor
(including left/right module structures induced on each,
left/right exactness, behavior of projectives/injectives,
flatness, and the adjoint isomorphism)
II. Noncommutative Ring Theory
Noetherian, Artinian, Hilbert Basis Theorem
Composition Series, Length
Semisimple Rings and Modules
Maschke's Theorem
Schur's Lemma
ArtinWedderburn Theorem
Jacobson Radical, Nakayama's Lemma
Artinian Rings
KrullSchmidt Theorem
III. Additional Commutative Topic
Localisation


731 SOURCES

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 Spring '11
 NA
 Algebra, Vector Space, Ring, Hilbert basis theorem, Joseph J. Rotman, Artinian Rings

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