2 % Abstract algebraic structures.
4 \ifx\havemjoalgebra\undefined
8 \ifx\operatorname\undefined
12 \input{mjo-common
} % for \of, and \binopmany
15 % The multiplicative identity element of its argument, which should be
16 % an algebraic structure.
17 \newcommand*
{\unit}[1]{ 1_
{{#1}} }
19 \ifdefined\newglossaryentry
20 \newglossaryentry{unit
}{
21 name=
{\ensuremath{\unit{R
}}},
22 description=
{the multiplicative identity (unit) element of $R$
},
27 % The direct sum of two things.
28 \newcommand*
{\directsum}[2]{ {#1}\oplus{#2} }
30 % The direct sum of three things.
31 \newcommand*
{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
33 % The (indexed) direct sum of many things.
34 \newcommand*
{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
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 $X$
},
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}}}
56 % The ideal generated by its argument, a subset consisting of ring or
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 $X$
},
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 $X$
},