Quotient Rings

Quotient rings $Q = R/I$ in Singular.jl are constructed with the constructor QuotientRing(R, I). The input ideal $I$ to the constructor must be a Groebner basis. The $R$-ideal $I$ may be recovered as quotient_ideal(Q).

Singular.is_quotient_ring — Method

is_quotient_ring(R::PolyRingUnion)

Return true if the given ring is the quotient of a polynomial ring with a non - zero ideal.

source

Singular.quotient_ideal — Method

quotient_ideal(Q::PolyRing{T}) where T <: Nemo.RingElem

Return I for a given quotient ring Q = R/I.

source

Examples

julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);

julia> is_quotient_ring(R)
false

julia> Q1, (x, y) = QuotientRing(R, Ideal(R, x^2+y^2));

julia> is_quotient_ring(Q1)
true

julia> quotient_ideal(Q1)
Singular ideal over Singular polynomial ring (QQ),(x,y),(dp(2),C) with generators (x^2 + y^2)

julia> Q2, (x, y) = QuotientRing(Q1, std(Ideal(Q1, x*y)));

julia> quotient_ideal(Q2)
Singular ideal over Singular polynomial ring (QQ),(x,y),(dp(2),C) with generators (x*y, y^3, x^2 + y^2)

julia> base_ring(quotient_ideal(Q1)) == base_ring(quotient_ideal(Q2))
true