Equality and containment between two ideals in a polynomial ring (or quotient of a polynomial ring) is checked by comparing their respective Groebner bases.
equal and not equal
Use
Ideal == Ideal to test if two ideals in the same ring are equal.
i1 : R = QQ[a..d];
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i2 : I = ideal (a^2*b-c^2, a*b^2-d^3, c^5-d);
o2 : Ideal of R
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i3 : J = ideal (a^2,b^2,c^2,d^2);
o3 : Ideal of R
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i4 : I == J
o4 = false
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i5 : I != J
o5 = true
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normal form with respect to a Groebner basis and membership
The function
RingElement % Ideal reduces an element with respect to a Groebner basis of the ideal.
i6 : (1+a+a^3+a^4) % J
o6 = a + 1
o6 : R
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We can then test membership in the ideal by comparing the answer to 0 using
==.
i7 : (1+a+a^3+a^4) % J == 0
o7 = false
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i8 : a^4 % J == 0
o8 = true
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containment for two ideals
Containment for two ideals is tested using
isSubset.
i9 : isSubset(I,J)
o9 = false
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i10 : isSubset(I,I+J)
o10 = true
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i11 : isSubset(I+J,I)
o11 = false
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ideal equal to 1 or 0
Use the expression
I == 1 to see if the ideal is equal to the ring. Use
I == 0 to see if the ideal is identically zero in the given ring.
i12 : I = ideal (a^2-1,a^3+3);
o12 : Ideal of R
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i13 : I == 1
o13 = true
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i14 : S = R/I
o14 = S
o14 : QuotientRing
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i15 : S == 0
o15 = true
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