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 | 20x+40y -39x-26y 6x-34y 34x+47y -x-21y -31x+14y -19x+40y -9x+28y |
| -40x+35y -33x-22y -19x-46y 14x-39y -20x+8y -13x-48y -46x-17y 8x+9y |
| -44x-49y 23x+50y -22x+50y -2x+11y 3x-13y 13x-3y 33x+32y 30x-49y |
| 42x+35y -5x+29y 48x+41y -30x-36y -27x-35y 18x+25y -46x+10y -46x+4y |
| 11x+50y 34x+23y 29x+25y -47x-33y 19x-19y -47x+36y 45y 3x+30y |
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 |, | 46 -39 26 17 -40 |)
| 0 0 x 0 y 0 0 0 | | 2 1 1 24 50 |
| 0 0 0 y x 0 0 0 | | -24 -49 0 3 3 |
| 0 0 0 0 0 x 0 y | | 1 0 0 0 0 |
| 0 0 0 0 0 0 y x | | -50 7 8 -4 -6 |
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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