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 | -x+37y 18x+8y -42x+49y 25x-13y 19x-47y 8x -26x+40y 48x+20y |
| -2x+47y -9x+32y 15x-36y 42x-12y -40x-4y 14x-29y 21x-33y -3x-41y |
| -25x+11y -7x-44y 24x+12y 25x-6y 25x-8y -26x-12y 43x+42y 28x-37y |
| -13x-47y 38x+32y 26x+19y -46x+29y -13x+13y 45x+31y 35x-y 5x-12y |
| 13x-36y 12x-37y -43x-11y 15x-19y -43x-14y -9x-14y 48x-46y -22x-5y |
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 |, | -10 -20 29 -38 32 |)
| 0 0 x 0 y 0 0 0 | | -41 19 -34 16 22 |
| 0 0 0 y x 0 0 0 | | -36 -36 9 4 -30 |
| 0 0 0 0 0 x 0 y | | 13 21 31 -10 30 |
| 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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