This function decomposes a module into a direct sum of simple modules, given some fairly strong assumptions on the ring which acts on the ring which acts on the module. This ring must only have two variables, and the square of each of those variables must kill the module.
i1 : Q = ZZ/101[x,y]
o1 = Q
o1 : PolynomialRing
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i2 : R = Q/(x^2,y^2)
o2 = R
o2 : QuotientRing
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i3 : M = coker random(R^5, R^8 ** R^{-1})
o3 = cokernel | 24x-3y -34x-y -36x+y -10x+32y 50x-30y 43x-39y 2x-41y 46x-17y |
| 28x+31y -x-43y -34x+13y 46x+15y -31x+37y 5x-12y -23x+26y 35x-19y |
| -14x+44y -24x+26y -35x+26y 4x+30y 37x+33y 26x-9y -21x+28y 35x-17y |
| -x-37y 42x-4y 12x-25y -6x+37y -9x+10y 19x+41y -26x-32y 43x-49y |
| 40x-39y 37x-7y -15x-47y 32x-27y -45x-39y 5x+45y 42x+9y 26x+11y |
5
o3 : R-module, quotient of R
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i4 : (N,f) = decomposeModule M
o4 = (cokernel | y x 0 0 0 0 0 0 |, | -11 31 -1 35 18 |)
| 0 0 x 0 y 0 0 0 | | 50 39 -9 -36 -42 |
| 0 0 0 y x 0 0 0 | | 17 -30 35 -36 3 |
| 0 0 0 0 0 x 0 y | | 1 0 0 0 0 |
| 0 0 0 0 0 0 y x | | 50 -48 -24 -34 -23 |
o4 : Sequence
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i5 : components N
o5 = {cokernel | y x |, cokernel | x 0 y |, cokernel | x 0 y |}
| 0 y x | | 0 y x |
o5 : List
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i6 : ker f == 0
o6 = true
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i7 : coker f == 0
o7 = true
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