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 | 8x+25y 19x-36y -42x+11y 8x+7y 23x-19y 43x-8y 44x-10y -38x-47y |
| 26y 8x-42y -11x+6y 20x-38y -31x-34y -27x-14y 47x-7y -15x-38y |
| -46x+50y 21x-35y 20x-7y 4x-3y -6x-36y 48x+5y -43x+21y 38x-40y |
| 49x+24y 35x -13x+y -16x+42y -9x+27y 22x+12y -25x-8y -34x+10y |
| -43x+21y -8x+26y -18x+19y 7x+37y 46x+40y -8x+21y 44x+5y -47x+6y |
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 |, | -5 -26 -44 1 -21 |)
| 0 0 x 0 y 0 0 0 | | 19 19 13 -5 -2 |
| 0 0 0 y x 0 0 0 | | 0 31 -21 19 31 |
| 0 0 0 0 0 x 0 y | | -37 -14 24 31 46 |
| 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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