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CoCoA starts with the default ring R = QQ[t,x,y,z]. The command for
building a new ring is:
I ::= C[X:INDETS]
or
I ::= C[X:INDETS], M_1, ... , M_n
where I is the identifier of a CoCoAL variable, C: RING is an
expression defining a coefficient ring (Z, Q or ZZ/(N)), X is an
expression that defines the indeterminates, and the
M_i are
modifiers. Each of these components is discussed in separate
sections, below.
The modifiers are used to modify the default settings of the base
ring. The modifiers are of three classes: termorderings, weights,
term orderings for modules. These classes are discussed in separate
sections below. It is possible to have no modifiers. The default
values are: DegRevLex for the termordering, 1 for the weight of each
indeterminate, and ToPos for the module termordering.
After the ring is defined using the above syntax, it can be made to be
the current ring with the command
Use
or it can be accessed
temporarily from within the current ring with the command
Using
. See these two commands for more information. There are also
several examples of ringbuilding in the tutorial.
Use R ::= QQ[x,y,z];  define and use the ring R
S ::= ZZ/(103)[x,y], Lex;  define the ring S with Lex termordering
CurrentRing();  the current ring is still R
QQ[x,y,z]

Use S;  now the ring S is the current ring
ZZ/(103)[x,y]

Using R Do X := z^23 EndUsing;  define a variable in R (not the current
 ring)
Y := R::x^2+y^2+z^2;  another way of defining a variable in R while
 S is the current ring
T ::= QQ[x[1..4],y[1..5,1..5]], Elim(y[1..5,1]), ToPos;  a more
 complicated ring
