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 | -45x-y -2x-11y -48x+18y -32x+49y -23x-8y -7x-6y 27x-22y 13x+38y |
| 49x+20y -4x-45y -10x-22y -x+20y 47x-26y -7x+32y -37x-31y 50x+21y |
| 31x+20y -30x-50y -11x+6y -21x-6y 17x-48y -16x-4y -21x+18y -30x-36y |
| 38x+3y 39x+15y -4x+26y 11x-30y 40x+y 15x-29y -8x+24y 9x-27y |
| 13x-29y -35x-y -13x+6y 19x-25y -43x+10y 37x+9y 5x+41y -29x-24y |
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 |, | 25 -19 15 35 -10 |)
| 0 0 x 0 y 0 0 0 | | -3 -33 24 9 4 |
| 0 0 0 y x 0 0 0 | | -39 10 35 50 8 |
| 0 0 0 0 0 x 0 y | | 5 44 48 32 -6 |
| 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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