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
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i2 : A = QQ[u,v];
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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
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i4 : B = A/I;
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i5 : use A;
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i6 : C = A/(u^2-v^2,u*v);
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i7 : D = GF(9,Variable=>a)[x,y]/(y^2 - x*(x-1)*(x-a))
o7 = D
o7 : QuotientRing
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i8 : ambient D
o8 = GF 9[x..y]
o8 : PolynomialRing
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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
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i10 : R = ZZ/101[t]
o10 = R
o10 : PolynomialRing
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i11 : R/t === R/t
o11 = false
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