Intro to Modules

\(\DeclareMathOperator{\ann}{ann} \DeclareMathOperator{\Ass}{Ass} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\can}{can} \DeclareMathOperator{\Ch}{Ch} \DeclareMathOperator{\char}{char} \DeclareMathOperator{\cp}{CharPoly} \DeclareMathOperator{\codim}{codim} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\colim}{colim} \DeclareMathOperator{\cone}{cone} \DeclareMathOperator{\cont}{cont} \DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\dim}{dim} \DeclareMathOperator{\edim}{embdim} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\eval}{eval} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Frac}{Frac} \DeclareMathOperator{\Fun}{Fun} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\gcd}{gcd} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\height}{height} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Inn}{Inn} \DeclareMathOperator{\ker}{ker} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\max}{max} \DeclareMathOperator{\Min}{Min} \DeclareMathOperator{\mp}{MinPoly} \DeclareMathOperator{\mSpec}{mSpec} \DeclareMathOperator{\Nul}{Nul} \DeclareMathOperator{\ns}{nullspace} \DeclareMathOperator{\opp}{opp} \DeclareMathOperator{\Orb}{Orb} \DeclareMathOperator{\Out}{Out} \DeclareMathOperator{\pdim}{pdim} \DeclareMathOperator{\Perm}{Perm} \DeclareMathOperator{\PtStab}{pStab} \DeclareMathOperator{\rad}{rad} \DeclareMathOperator{\range}{range} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\res}{res} \DeclareMathOperator{\sStab}{SetStab} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\Span}{Span} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Stab}{Stab} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Syl}{Syl} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\trace}{trace} \DeclareMathOperator{\trdeg}{trdeg} \newcommand{\Ob}{\mathrm{Ob}} 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Chapter 11 Intro to Modules

Motivation: The What, Where, and Why of Modules.

For those of us who remember at least a little linear algebra, modules are generalizations of a vector spaces, but instead of using scalars from a field (like real or complex numbers), we allow scalars to come from a ring (which can be more complicated, like integers or polynomials). In short, modules are like "vector spaces with more structure" that allow us to work with rings instead of just fields, unlocking deeper connections in algebra, number theory, and beyond.

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