2 % Abstract algebraic structures.
4 \ifx\havemjoalgebra\undefined
8 \ifx\operatorname\undefined
12 \input{mjo-common
} % for \of, and \binopmany
15 % The additive identity element of its argument, which should be
16 % an algebraic structure.
17 \newcommand*
{\zero}[1]{ 0_
{{#1}} }
19 \ifdefined\newglossaryentry
20 \newglossaryentry{zero
}{
21 name=
{\ensuremath{\zero{R
}}},
22 description=
{the additive identity element of $R$
},
27 % The multiplicative identity element of its argument, which should be
28 % an algebraic structure.
29 \newcommand*
{\unit}[1]{ 1_
{{#1}} }
31 \ifdefined\newglossaryentry
32 \newglossaryentry{unit
}{
33 name=
{\ensuremath{\unit{R
}}},
34 description=
{the multiplicative identity (unit) element of $R$
},
39 % The direct sum of two things.
40 \newcommand*
{\directsum}[2]{ {#1}\oplus{#2} }
42 % The direct sum of three things.
43 \newcommand*
{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
45 % The (indexed) direct sum of many things.
46 \newcommand*
{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
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 $X$
},
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}}}
68 % The ideal generated by its argument, a subset consisting of ring or
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 $X$
},
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 $X$
},
93 % The stabilizer subgroup of its first argument that fixes the point
94 % given by its second argument.
95 \newcommand*
{\Stab}[2]{ #1_
{#2} }