mjo-hurwitz.tex: new file for Hurwitz algebras (quaternions and octonions).
[mjotex.git] / 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 additive identity element of its argument, which should be
16 % an algebraic structure.
17 \newcommand*{\zero}[1]{ 0_{{#1}} }
18
19 \ifdefined\newglossaryentry
20 \newglossaryentry{zero}{
21 name={\ensuremath{\zero{R}}},
22 description={the additive identity element of $R$},
23 sort=z
24 }
25 \fi
26
27 % The multiplicative identity element of its argument, which should be
28 % an algebraic structure.
29 \newcommand*{\unit}[1]{ 1_{{#1}} }
30
31 \ifdefined\newglossaryentry
32 \newglossaryentry{unit}{
33 name={\ensuremath{\unit{R}}},
34 description={the multiplicative identity (unit) element of $R$},
35 sort=u
36 }
37 \fi
38
39 % The direct sum of two things.
40 \newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
41
42 % The direct sum of three things.
43 \newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
44
45 % The (indexed) direct sum of many things.
46 \newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
47
48
49 % The (sub)algebra generated by its argument, a subset of some ambient
50 % algebra. By definition this is the smallest subalgebra (of the
51 % ambient one) containing that set.
52 \newcommand*{\alg}[1]{\operatorname{alg}\of{{#1}}}
53 \ifdefined\newglossaryentry
54 \newglossaryentry{alg}{
55 name={\ensuremath{\alg{X}}},
56 description={the (sub)algebra generated by $X$},
57 sort=a
58 }
59 \fi
60
61
62 % The fraction field of its argument, an integral domain. The name
63 % "Frac" was chosen here instead of "Quot" because the latter
64 % corresponds to the term "quotient field," which can be mistaken in
65 % some cases for... a quotient field (something mod something).
66 \newcommand*{\Frac}[1]{\operatorname{Frac}\of{{#1}}}
67
68 % The ideal generated by its argument, a subset consisting of ring or
69 % algebra elements.
70 \newcommand*{\ideal}[1]{\operatorname{ideal}\of{{#1}}}
71 \ifdefined\newglossaryentry
72 \newglossaryentry{ideal}{
73 name={\ensuremath{\ideal{X}}},
74 description={the ideal generated by $X$},
75 sort=i
76 }
77 \fi
78
79
80 % The polynomial ring whose underlying commutative ring of
81 % coefficients is the first argument and whose indeterminates (a
82 % comma-separated list) are the second argumnt.
83 \newcommand*{\polyring}[2]{{#1}\left[{#2}\right]}
84 \ifdefined\newglossaryentry
85 \newglossaryentry{polyring}{
86 name={\ensuremath{\polyring{R}{X}}},
87 description={polynomials with coefficients in $R$ and variable $X$},
88 sort=p
89 }
90 \fi
91
92
93 \fi