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} % for \of, and \binopmany
  13
  14
  15 % The direct sum of two things.
  16 \newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
  17
  18 % The direct sum of three things.
  19 \newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
  20
  21 % The (indexed) direct sum of many things.
  22 \newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
  23
  24
  25 % The (sub)algebra generated by its argument, a subset of some ambient
  26 % algebra. By definition this is the smallest subalgebra (of the
  27 % ambient one) containing that set.
  28 \newcommand*{\alg}[1]{\operatorname{alg}\of{{#1}}}
  29 \ifdefined\newglossaryentry
  30   \newglossaryentry{alg}{
  31     name={\ensuremath{\alg{X}}},
  32     description={the (sub)algebra generated by $X$},
  33     sort=a
  34   }
  35 \fi
  36
  37
  38 % The fraction field of its argument, an integral domain. The name
  39 % "Frac" was chosen here instead of "Quot" because the latter
  40 % corresponds to the term "quotient field," which can be mistaken in
  41 % some cases for... a quotient field (something mod something).
  42 \newcommand*{\Frac}[1]{\operatorname{Frac}\of{{#1}}}
  43
  44 % The ideal generated by its argument, a subset consisting of ring or
  45 % algebra elements.
  46 \newcommand*{\ideal}[1]{\operatorname{ideal}\of{{#1}}}
  47 \ifdefined\newglossaryentry
  48   \newglossaryentry{ideal}{
  49     name={\ensuremath{\ideal{X}}},
  50     description={the ideal generated by $X$},
  51     sort=i
  52   }
  53 \fi
  54
  55
  56 % The polynomial ring whose underlying commutative ring of
  57 % coefficients is the first argument and whose indeterminates (a
  58 % comma-separated list) are the second argumnt.
  59 \newcommand*{\polyring}[2]{{#1}\left[{#2}\right]}
  60
  61
  62 \fi