\contentsline {part}{I\hspace {1em}Fundamentals}{1}{}%
\contentsline {chapter}{\numberline {1}Rings}{5}{}%
\contentsline {section}{\numberline {1.1}Rings}{5}{}%
\contentsline {section}{\numberline {1.2}Properties of rings}{6}{}%
\contentsline {section}{\numberline {1.3}Zero divisors and units}{7}{}%
\contentsline {section}{\numberline {1.4}Domains}{7}{}%
\contentsline {section}{\numberline {1.5}Fields}{8}{}%
\contentsline {subsection}{\numberline {1.5.1}Field extension}{8}{}%
\contentsline {subsection}{\numberline {1.5.2}Algebraic extension}{8}{}%
\contentsline {chapter}{\numberline {2}Quotient rings}{9}{}%
\contentsline {section}{\numberline {2.1}Ideals}{9}{}%
\contentsline {section}{\numberline {2.2}Ideal generating notation}{11}{}%
\contentsline {section}{\numberline {2.3}Basic types of ideals}{12}{}%
\contentsline {subsection}{\numberline {2.3.1}Zero ideals}{12}{}%
\contentsline {subsection}{\numberline {2.3.2}Unit ideals}{12}{}%
\contentsline {subsection}{\numberline {2.3.3}Principle ideal}{13}{}%
\contentsline {section}{\numberline {2.4}Quotient rings}{13}{}%
\contentsline {subsection}{\numberline {2.4.1}Examples of quotient rings}{13}{}%
\contentsline {section}{\numberline {2.5}Prime ideal}{13}{}%
\contentsline {section}{\numberline {2.6}Maximal ideal}{14}{}%
\contentsline {section}{\numberline {2.7}Generalized number theory in integral domains}{15}{}%
\contentsline {chapter}{\numberline {3}Ring homomorphisms}{17}{}%
\contentsline {section}{\numberline {3.1}Ring homomorphism}{17}{}%
\contentsline {section}{\numberline {3.2}Properties of ring homomorphism}{18}{}%
\contentsline {section}{\numberline {3.3}Examples of ring homomorphism}{18}{}%
\contentsline {section}{\numberline {3.4}Ring monomorphism}{18}{}%
\contentsline {section}{\numberline {3.5}Ring epimorphism}{18}{}%
\contentsline {section}{\numberline {3.6}Ring endomorphism}{18}{}%
\contentsline {chapter}{\numberline {4}Ring isomorphism}{19}{}%
\contentsline {section}{\numberline {4.1}Properties of Ring isomorphism}{19}{}%
\contentsline {section}{\numberline {4.2}Examples of Ring isomorphism}{19}{}%
\contentsline {section}{\numberline {4.3}Ring automorphism}{19}{}%
\contentsline {section}{\numberline {4.4}First isomorphism theorem}{20}{}%
\contentsline {section}{\numberline {4.5}Ring characteristic}{21}{}%
\contentsline {subsection}{\numberline {4.5.1}Theorems related to this special homomorphism}{22}{}%
\contentsline {subsection}{\numberline {4.5.2}Frobenius map}{22}{}%
\contentsline {chapter}{\numberline {5}Field of fractions}{23}{}%
\contentsline {section}{\numberline {5.1}Divisibility in a domain}{24}{}%
\contentsline {section}{\numberline {5.2}GCD in a domain}{24}{}%
\contentsline {chapter}{\numberline {6}Associative algebra}{25}{}%
\contentsline {chapter}{\numberline {7}Polynomial rings}{27}{}%
\contentsline {chapter}{\numberline {8}Product Rings}{29}{}%
\contentsline {chapter}{\numberline {9}Modules}{31}{}%
\contentsline {part}{II\hspace {1em}Advanced}{33}{}%
\contentsline {chapter}{\numberline {10}Torsion elements}{35}{}%
\contentsline {section}{\numberline {10.1}Annihilators}{35}{}%