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 | -7x+45y -31x-49y 11x+21y 14x+35y -47x-8y -3x-27y 7x+9y 19x-17y |
| 21x+25y -26x+45y 13x-41y -46x+17y -30x-8y -14x-48y 35x+47y 27x-18y |
| 30x+33y -45x-17y -2x+29y 23x+15y 27x-50y 44x+24y 9x+46y 18x+44y |
| 31x+4y -22x+17y -50x-22y -5x-45y -32x-7y 26x-39y 17x-32y -22x+33y |
| -17y 47x-49y -40x-23y 34x-33y -x-13y -10x-6y 4x+11y 25x+6y |
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 |, | 34 -33 25 29 46 |)
| 0 0 x 0 y 0 0 0 | | -21 5 -41 -40 -10 |
| 0 0 0 y x 0 0 0 | | -18 12 -16 37 47 |
| 0 0 0 0 0 x 0 y | | 2 -7 31 -50 4 |
| 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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