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 | -11x-46y -48x+11y -35x-42y 4x+12y -27x+33y 18x+5y -48x+16y 38x+23y |
| 44x+9y 34x-40y 48x+23y -41x+2y 23x+24y -7x-24y 38x-43y 37x+23y |
| -41x-5y 19x-5y -20x+50y 10x-24y -4x-32y -44x-27y 12x-37y 39x-50y |
| 4x+6y 3x+46y -30x-15y -17x+50y 24x-17y -15x+22y -11x+37y 33x-22y |
| -35x+9y -24x-42y 49x+49y 18y -2x+48y -28x+6y -23x+46y 48x-18y |
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 |, | -6 22 -33 17 -30 |)
| 0 0 x 0 y 0 0 0 | | 34 -32 -46 9 -46 |
| 0 0 0 y x 0 0 0 | | -5 21 -32 -3 -48 |
| 0 0 0 0 0 x 0 y | | 1 0 0 0 0 |
| 0 0 0 0 0 0 y x | | -33 -39 -7 -16 31 |
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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