quotient -- ideal or submodule quotient

quotient -- ideal or submodule quotient

quotient(I,J) -- computes the ideal or submodule quotient (I:J).

The arguments should be ideals in the same ring, or submodules of the same module. If J is a ring element, then the principal ideal generated by J is used.

The operator : can be used as an abbreviation, but without optional arguments; see Module : Module.

For ideals, the quotient is the set of ring elements r such that rJ is contained in I. If I is a submodule of a module M, and J is an ideal, the quotient is the set of elements m of M such that Jm is contained in I. Finally, if I and J are submodules of the same module M, then the result is the set of ring elements r such that rJ is contained in I.

i1 : R = ZZ/32003[a..d];

i2 : J = monomialCurveIdeal(R,{1,4,7})



             2         3     3    4    3

o2 = ideal (c  - b*d, b c - a d, b  - a c)



o2 : Ideal of R

i3 : I = ideal(J_1-a^2*J_0,J_2-d*c*J_0)



             3     2 2    3     2      4    3     3         2

o3 = ideal (b c - a c  - a d + a b*d, b  - a c - c d + b*c*d )



o3 : Ideal of R

i4 : I : J



             3    2       2   3    2     2

o4 = ideal (b  - a c + c*d , a  - a b + c d)



o4 : Ideal of R

The computation is currently not stored anywhere: this means that the computation cannot be continued after an interrupt. This will be changed in a later version.

Ways to use quotient :

quotient(Ideal,Ideal)

quotient(Ideal,RingElement)

quotient(Module,Ideal)

quotient(Module,Module)

quotient(Module,RingElement)

Optional arguments :

quotient(..., MinimalGenerators => ...)

quotient(..., Strategy => ...) -- specify a computational strategy