2 % Abstract algebraic structures.
4 \ifx\havemjoalgebra\undefined
8 \ifx\operatorname\undefined
12 \input{mjo-common
} % for \of, and \binopmany
15 % The direct sum of two things.
16 \newcommand*
{\directsum}[2]{ {#1}\oplus{#2} }
18 % The direct sum of three things.
19 \newcommand*
{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
21 % The (indexed) direct sum of many things.
22 \newcommand*
{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
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 $X$
},
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}}}
44 % The ideal generated by its argument, a subset consisting of ring or
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 $X$
},
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]}