S,ε = polynomial_ring(QQ,"ε")
S,pi = quo(S,ε^2)
R,(x1,x2,x3) = S["x1","x2","x3"]
intersect(ideal([-ε*x1*x2 + x1*x3 - x2*x3 + x3^2]),ideal([x1^2*x2*x3 - x1*x2^2*x3 + x1*x2*x3^2]))

julia> intersect(ideal([-ε*x1*x2 + x1*x3 - x2*x3 + x3^2]),ideal([x1^2*x2*x3 - x1*x2^2*x3 + x1*x2*x3^2]))
ERROR: function divrem is not implemented for arguments
EuclideanRingResidueRingElem{QQPolyRingElem}: ε
EuclideanRingResidueRingElem{QQPolyRingElem}: -ε

Stacktrace:
 [1] divrem(a::EuclideanRingResidueRingElem{QQPolyRingElem}, b::EuclideanRingResidueRingElem{QQPolyRingElem})
   @ AbstractAlgebra ~/.julia/packages/AbstractAlgebra/AqzuZ/src/algorithms/GenericFunctions.jl:50
 [2] divrem(a::EuclideanRingResidueRingElem{QQPolyRingElem}, b::EuclideanRingResidueRingElem{QQPolyRingElem})
   @ AbstractAlgebra ~/.julia/packages/AbstractAlgebra/AqzuZ/src/AbstractAlgebra.jl:65
 [3] gcdx(a::EuclideanRingResidueRingElem{QQPolyRingElem}, b::EuclideanRingResidueRingElem{QQPolyRingElem})
   @ AbstractAlgebra ~/.julia/packages/AbstractAlgebra/AqzuZ/src/algorithms/GenericFunctions.jl:306
 [4] nemoRingExtGcd(a::Ptr{Nothing}, b::Ptr{Nothing}, s::Ptr{Ptr{Nothing}}, t::Ptr{Ptr{Nothing}}, cf::Ptr{Nothing})
   @ Singular.libSingular ~/.julia/packages/Singular/JyB5B/src/libsingular/nemo/Rings.jl:200
 [5] id_Intersection
   @ ~/.julia/packages/CxxWrap/eWADG/src/CxxWrap.jl:668 [inlined]
 [6] intersection(I::Singular.sideal{Singular.spoly{…}}, J::Singular.sideal{Singular.spoly{…}})
   @ Singular ~/.julia/packages/Singular/JyB5B/src/ideal/ideal.jl:508
 [7] intersect(I::MPolyIdeal{AbstractAlgebra.Generic.MPoly{…}}, Js::MPolyIdeal{AbstractAlgebra.Generic.MPoly{…}})
   @ Oscar ~/.julia/dev/Oscar/src/Rings/mpoly-ideals.jl:217
 [8] top-level scope
   @ REPL[265]:1
Some type information was truncated. Use `show(err)` to see complete types.

R,(x1,x2) = polynomial_ring(QQ,["x1","x2"])
x1//2 == x1/2 # true
phi = hom(R,R,c->c,gens(R))
phi(x1//2) # does not work
phi(x1/2)  # works

julia> R,(x1,x2) = polynomial_ring(QQ,["x1","x2"])
(Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x1, x2])

julia> x1//2 == x1/2
true

julia> phi = hom(R,R,c->c,gens(R))
Ring homomorphism
  from multivariate polynomial ring in 2 variables over QQ
  to multivariate polynomial ring in 2 variables over QQ
defined by
  x1 -> x1
  x2 -> x2
with map on coefficients
  #41

julia> phi(x1//2) # does not work
ERROR: MethodError: no method matching (::QQMPolyRing)(::AbstractAlgebra.Generic.FracFieldElem{QQMPolyRingElem})

Closest candidates are:
  (::QQMPolyRing)(::Vector{QQFieldElem}, ::Vector{Vector{Int64}})
   @ Nemo ~/.julia/packages/Nemo/pX5Oz/src/flint/fmpq_mpoly.jl:1236
  (::QQMPolyRing)(::Vector{Any}, ::Array{Vector{T}, 1}) where T
   @ Nemo ~/.julia/packages/Nemo/pX5Oz/src/flint/fmpq_mpoly.jl:1248
  (::QQMPolyRing)(::Vector{QQFieldElem}, ::Array{Vector{T}, 1}) where T<:Union{UInt64, ZZRingElem}
   @ Nemo ~/.julia/packages/Nemo/pX5Oz/src/flint/fmpq_mpoly.jl:1224
  ...

Stacktrace:
 [1] (::Oscar.MPolyAnyMap{…})(g::AbstractAlgebra.Generic.FracFieldElem{…})
   @ Oscar ~/.julia/dev/Oscar/src/Rings/MPolyMap/MPolyRing.jl:179
 [2] top-level scope
   @ REPL[156]:1
Some type information was truncated. Use `show(err)` to see complete types.

julia> phi(x1/2)  # works
1//2*x1