next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
Macaulay2Doc :: Ring / Ideal

Ring / Ideal -- make a quotient ring

Synopsis

Description

If I is a ring element of R or ZZ, or a list or sequence of such elements, then I is understood to be the ideal generated by these elements. If I is a module, then it must be a submodule of a free module of rank 1.
i1 : ZZ[x]/367236427846278621

            ZZ[x]
o1 = ------------------
     367236427846278621

o1 : QuotientRing
i2 : A = QQ[u,v];
i3 : I = ideal random(A^1, A^{-2,-2,-2})

            9 2   1      9 2  1 2         3 2  3 2   3      7 2
o3 = ideal (-u  + -u*v + -v , -u  + u*v + -v , -u  + -u*v + -v )
            2     2      4    2           4    2     4      4

o3 : Ideal of A
i4 : B = A/I;
i5 : use A;
i6 : C = A/(u^2-v^2,u*v);
i7 : D = GF(9,Variable=>a)[x,y]/(y^2 - x*(x-1)*(x-a))

o7 = D

o7 : QuotientRing
i8 : ambient D

o8 = GF 9[x, y]

o8 : PolynomialRing
The names of the variables are assigned values in the new quotient ring (by automatically running use R) when the new ring is assigned to a global variable.

Warning: quotient rings are bulky objects, because they contain a Gröbner basis for their ideals, so only quotients of ZZ are remembered forever. Typically the ring created by R/I will be a brand new ring, and its elements will be incompatible with the elements of previously created quotient rings for the same ideal.

i9 : ZZ/2 === ZZ/(4,6)

o9 = true
i10 : R = ZZ/101[t]

o10 = R

o10 : PolynomialRing
i11 : R/t === R/t

o11 = false