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 | -49x+23y x-6y -28x-46y 38x+16y x+4y 5x+17y 39x-14y -36x+8y |
| -40x-43y 7x-34y -29x+9y -27x+16y -4x-34y 14x-23y -50x-22y -13x-45y |
| 30x+32y -11x-9y -3x+6y 4x-28y -15x-20y 33x-45y -9x-11y 21x-40y |
| -24x+36y 43x+19y -7x+47y -28x-35y 39x+33y 24x-7y 13x+21y -29x-15y |
| -8x-11y -14x-10y 20x-41y 42x-48y 16x+23y 19x-36y 2x-4y 23x+32y |
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 |, | 20 -24 2 46 14 |)
| 0 0 x 0 y 0 0 0 | | -9 40 -19 34 -11 |
| 0 0 0 y x 0 0 0 | | 6 23 -16 -36 15 |
| 0 0 0 0 0 x 0 y | | 44 15 -32 -18 27 |
| 0 0 0 0 0 0 y x | | 1 0 0 0 0 |
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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