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 | -27x-47y -2x-9y 23x-27y 27x+33y -21x+13y 31x-47y 19x+3y 28x+37y |
| -39x+14y -2x+26y -49x-30y -8y -23x+17y 26x-12y -26x+8y 39x-39y |
| 21x+34y -24x+22y x+25y 26x-20y -30x+4y -31y -x+40y -9x-37y |
| -40x+49y -35x+45y -38x+46y 38x+27y -12x+42y 20x-49y -34x-35y -43x-8y |
| -25x-32y 39x-11y 31x+28y 24x-48y x+10y 8x+6y 31x-19y 5x-45y |
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 |, | 27 -21 41 5 -31 |)
| 0 0 x 0 y 0 0 0 | | 10 12 50 5 10 |
| 0 0 0 y x 0 0 0 | | 27 6 10 -29 -1 |
| 0 0 0 0 0 x 0 y | | 1 0 0 0 0 |
| 0 0 0 0 0 0 y x | | -16 -46 12 -49 -41 |
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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