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 | -43x-17y -18x-42y -47x+40y 18x-13y 38x+14y -19x-45y 46x+15y 35x-46y |
| 25x-39y -46x-5y -27x-34y 5x-15y -12x+38y -28x+11y 20x+41y -3x+42y |
| 50x-22y -38x+32y -6x-47y 44x-23y -21x+36y 12x-8y 14x+44y -17x+39y |
| 35x-20y 44x+49y x-43y 34x+27y -39x+19y 34x-31y 18x+2y -28x+3y |
| 18x-12y -7x-31y 22x+9y 48x+25y -32x-35y -45x+28y 2x+46y 9x+38y |
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 |, | 32 -45 46 5 34 |)
| 0 0 x 0 y 0 0 0 | | 16 -1 -37 19 -40 |
| 0 0 0 y x 0 0 0 | | 49 -1 20 36 20 |
| 0 0 0 0 0 x 0 y | | -11 -50 5 50 43 |
| 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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