Description
In order to check whether a matrix, whose source module is not free, is well defined, then a Gröbner basis computation will probably be required.
i1 : R = QQ[a..d];
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i2 : f = map(R^1,coker vars R,{{1_R}})
o2 = | 1 |
o2 : Matrix
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i3 : isWellDefined f
o3 = false
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i4 : isWellDefined map(coker vars R, R^1, {{1_R}})
o4 = true
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In order to check whether a ring map is well defined, it is often necessary to check that the image of an ideal under a related ring map is zero. This often requires a Gröbner basis as well.
i5 : A = ZZ/5[a]
o5 = A
o5 : PolynomialRing
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i6 : factor(a^3-a-2)
3
o6 = (a - a - 2)
o6 : Expression of class Product
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i7 : B = A/(a^3-a-2);
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i8 : isWellDefined map(A,B)
o8 = false
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i9 : isWellDefined map(B,A)
o9 = true
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