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 | 5x-32y 48x-29y 25x-46y 22x-2y 35x+17y 18x-35y 6x-32y -28x+23y |
| -47x+45y 33x-37y -36x+27y 24x-33y 39x-20y -5x-17y 2x-15y -31x-11y |
| 2x-33y -9x+20y -x+22y -4x+35y 35x+18y 47x-42y 46x-29y 33x-28y |
| 42x-6y -x-8y 29x-49y -x-27y 9x-46y 4x+19y -14x-16y 47x-23y |
| -42x-42y -38x+38y 38x-19y -24x-50y 14x+30y 13x-33y 5x+43y 11x+2y |
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 |, | 4 -23 -11 -19 -18 |)
| 0 0 x 0 y 0 0 0 | | 30 15 48 12 -14 |
| 0 0 0 y x 0 0 0 | | -50 4 -1 48 11 |
| 0 0 0 0 0 x 0 y | | 1 0 0 0 0 |
| 0 0 0 0 0 0 y x | | -33 11 30 39 27 |
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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