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 | -31x+14y 50x-16y 39x-28y 14x+23y 6x+50y 12x+50y -28x+3y -35x-26y |
| 8x-13y 17x-14y -4x+46y 27x-23y 20x+24y 13x+17y 48x-17y 27x-25y |
| 14x-30y 9x-27y -47x+43y -38x-21y -2x-37y 20x-31y 14x-25y -12x+38y |
| 39x-33y 6x-39y 31x+9y 10x-15y -16x+39y 28x-38y -11x-4y -36x+14y |
| 5x-38y -49x+17y -25x+7y 30x-42y 5x-50y 42x+20y -42x-37y -32x+36y |
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 |, | -7 -20 -13 -1 2 |)
| 0 0 x 0 y 0 0 0 | | 24 22 9 -2 2 |
| 0 0 0 y x 0 0 0 | | 27 33 -17 21 -42 |
| 0 0 0 0 0 x 0 y | | 1 0 0 0 0 |
| 0 0 0 0 0 0 y x | | -21 32 40 38 25 |
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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