mjo-algebra.tex

   1 %
   2 % Abstract algebraic structures.
   3 %
   4 \ifx\havemjoalgebra\undefined
   5 \def\havemjoalgebra{1}
   6
   7
   8 \ifx\operatorname\undefined
   9   \usepackage{amsopn}
  10 \fi
  11
  12 \input{mjo-common}
  13
  14
  15 % The (sub)algebra generated by its argument, a subset of some ambient
  16 % algebra. By definition this is the smallest subalgebra (of the
  17 % ambient one) containing that set.
  18 \newcommand*{\alg}[1]{\operatorname{alg}\of{{#1}}}
  19 \ifdefined\newglossaryentry
  20   \newglossaryentry{alg}{
  21     name={\ensuremath{\alg{X}}},
  22     description={the (sub)algebra generated by $X$},
  23     sort=a
  24   }
  25 \fi
  26
  27
  28 % The fraction field of its argument, an integral domain. The name
  29 % "Frac" was chosen here instead of "Quot" because the latter
  30 % corresponds to the term "quotient field," which can be mistaken in
  31 % some cases for... a quotient field (something mod something).
  32 \newcommand*{\Frac}[1]{\operatorname{Frac}\of{{#1}}}
  33
  34 % The ideal generated by its argument, a subset consisting of ring or
  35 % algebra elements.
  36 \newcommand*{\ideal}[1]{\operatorname{ideal}\of{{#1}}}
  37 \ifdefined\newglossaryentry
  38   \newglossaryentry{ideal}{
  39     name={\ensuremath{\ideal{X}}},
  40     description={the ideal generated by $X$},
  41     sort=i
  42   }
  43 \fi
  44
  45
  46 % The polynomial ring whose underlying commutative ring of
  47 % coefficients is the first argument and whose indeterminates (a
  48 % comma-separated list) are the second argumnt.
  49 \newcommand*{\polyring}[2]{{#1}\left[{#2}\right]}
  50
  51
  52 \fi