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 multiplicative identity element of its argument, which should be
  16 % an algebraic structure.
  17 \newcommand*{\unit}[1]{ 1_{{#1}} }
  18
  19 \ifdefined\newglossaryentry
  20   \newglossaryentry{unit}{
  21     name={\ensuremath{\unit{R}}},
  22     description={the multiplicative identity (unit) element of $R$},
  23     sort=u
  24   }
  25 \fi
  26
  27 % The direct sum of two things.
  28 \newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
  29
  30 % The direct sum of three things.
  31 \newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
  32
  33 % The (indexed) direct sum of many things.
  34 \newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
  35
  36
  37 % The (sub)algebra generated by its argument, a subset of some ambient
  38 % algebra. By definition this is the smallest subalgebra (of the
  39 % ambient one) containing that set.
  40 \newcommand*{\alg}[1]{\operatorname{alg}\of{{#1}}}
  41 \ifdefined\newglossaryentry
  42   \newglossaryentry{alg}{
  43     name={\ensuremath{\alg{X}}},
  44     description={the (sub)algebra generated by $X$},
  45     sort=a
  46   }
  47 \fi
  48
  49
  50 % The fraction field of its argument, an integral domain. The name
  51 % "Frac" was chosen here instead of "Quot" because the latter
  52 % corresponds to the term "quotient field," which can be mistaken in
  53 % some cases for... a quotient field (something mod something).
  54 \newcommand*{\Frac}[1]{\operatorname{Frac}\of{{#1}}}
  55
  56 % The ideal generated by its argument, a subset consisting of ring or
  57 % algebra elements.
  58 \newcommand*{\ideal}[1]{\operatorname{ideal}\of{{#1}}}
  59 \ifdefined\newglossaryentry
  60   \newglossaryentry{ideal}{
  61     name={\ensuremath{\ideal{X}}},
  62     description={the ideal generated by $X$},
  63     sort=i
  64   }
  65 \fi
  66
  67
  68 % The polynomial ring whose underlying commutative ring of
  69 % coefficients is the first argument and whose indeterminates (a
  70 % comma-separated list) are the second argumnt.
  71 \newcommand*{\polyring}[2]{{#1}\left[{#2}\right]}
  72 \ifdefined\newglossaryentry
  73   \newglossaryentry{polyring}{
  74     name={\ensuremath{\polyring{R}{X}}},
  75     description={polynomials with coefficients in $R$ and variable $X$},
  76     sort=p
  77   }
  78 \fi
  79
  80
  81 \fi