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 | -15x+15y 13x-18y -21x+29y x-28y 25x+26y 35x+39y -36x+19y 5x+8y |
| -41x-3y -18x-48y 25x+26y 27x-9y -12x+30y 47x+37y -35x+7y 41x+3y |
| -40x+31y 5x-45y -32x+15y -49x+5y -15x+3y -50x-23y -20x+28y -48x-26y |
| 39x+12y -18x+41y -38x-3y -37x+25y -35x+12y 45x-28y -31x+17y 22x-10y |
| 50x+44y -12x-22y -39x+21y -40x-49y -34x+43y -4x-27y 6x+5y 27x-17y |
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 |, | -14 -8 6 -4 -31 |)
| 0 0 x 0 y 0 0 0 | | 39 29 -29 -45 -43 |
| 0 0 0 y x 0 0 0 | | -47 -28 50 -47 -4 |
| 0 0 0 0 0 x 0 y | | 1 0 0 0 0 |
| 0 0 0 0 0 0 y x | | 27 -40 42 50 38 |
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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