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 | -43x+25y -45x-48y -35x-40y -35x+43y -40x+33y 30x+9y 19x-23y -20x+49y |
| -5x+40y -33x-y 11x+11y -19x-4y -44x-26y -10x-31y 7x-44y -12x+8y |
| -20x-41y 23x-21y 17x+48y -19x-22y -44x-3y 44x+47y -19x-16y -46x+24y |
| -20x+17y -45x-50y -20x-25y -35x-10y 11x+28y -25x-14y -19x+38y -15x+40y |
| 19x-18y 35x+48y 6x+2y 35x+39y -7x+3y -x-35y x+25y -35x+39y |
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 |, | 11 40 -2 49 -46 |)
| 0 0 x 0 y 0 0 0 | | -31 -3 -47 -36 -28 |
| 0 0 0 y x 0 0 0 | | 13 21 -23 -33 -9 |
| 0 0 0 0 0 x 0 y | | -14 -3 -20 47 -42 |
| 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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