Calculator inputs
Use the stacked page layout for reading, while the input grid becomes three columns on large screens, two on medium screens, and one on mobile.
Example data table
| Case | Ring | Inputs | Principal generator | Comment |
|---|---|---|---|---|
| 1 | ℤ | 84, 126, 210 | 42 | The ideal (84, 126, 210) equals (42). |
| 2 | ℤ/12ℤ | 8, 10 | 2 | The ideal contains exactly 6 distinct residues. |
| 3 | ℤ/18ℤ | 6, 9, 15 | 3 | The ideal is proper and generated by divisor 3. |
| 4 | ℤ | 0, 0, 0 | 0 | The generated ideal is the zero ideal. |
Formula used
For the integer ring ℤ: the ideal generated by integers a₁, a₂, …, aₖ is
(a₁, a₂, …, aₖ) = (gcd(a₁, a₂, …, aₖ))
The calculator also produces a Bézout relation of the form
d = x₁a₁ + x₂a₂ + … + xₖaₖ
where d = gcd(a₁, a₂, …, aₖ).
For the residue ring ℤ/nℤ: the generated ideal is
(a₁, a₂, …, aₖ) = (gcd(n, a₁, a₂, …, aₖ))
If d = gcd(n, a₁, …, aₖ), then the ideal has n / d distinct residues.
Membership test: a value t belongs to (d) exactly when d | t in the normalized representative system.
Containment test: (d) ⊆ (c) when the comparison generator c divides d after normalization.
How to use this calculator
- Choose whether you want to work in the integer ring ℤ or the residue ring ℤ/nℤ.
- Enter one or more integer generators separated by commas, spaces, or line breaks.
- If you selected ℤ/nℤ, enter a positive modulus
n. - Add a membership test value to check whether a number lies inside the generated ideal.
- Add a comparison ideal generator to test containment against another principal ideal.
- Press Generate Principal Ideal to place the full result above the form, under the header.
- Review the generator, Bézout relation, gcd steps, sample elements, and graph.
- Use the CSV and PDF buttons to export the computed summary.
FAQs
1) What does this calculator actually generate?
It computes the principal ideal generated by your input set. In ℤ, that ideal is controlled by the gcd. In ℤ/nℤ, the modulus joins the gcd calculation, producing a divisor that determines the entire ideal.
2) Why is the gcd the ideal generator in ℤ?
Every linear combination of the inputs is divisible by the gcd, and Bézout’s identity shows the gcd itself is a linear combination. That makes the gcd both inside the ideal and strong enough to generate every other ideal element.
3) Why does the modulus matter in ℤ/nℤ?
In a residue ring, multiples are taken modulo n. The relevant generator becomes gcd(n, a₁, …, aₖ), because n itself is part of the algebraic structure and changes which residue classes can be produced.
4) What happens when every input is zero?
In ℤ, the result is the zero ideal (0), which contains only 0. In ℤ/nℤ, gcd(n, 0, …, 0) equals n, so the displayed divisor corresponds to the zero ideal, represented by the zero residue.
5) How is the membership test decided?
The calculator checks divisibility by the principal generator. In ℤ, a target lies in (d) when d divides it. In ℤ/nℤ, the target is first normalized modulo n, then checked against the same divisibility rule.
6) How is containment between two principal ideals tested?
For both supported settings, (d) is contained in (c) when c divides d after normalization. This follows the reverse divisibility order on principal ideals, where larger divisors usually generate smaller ideals.
7) Why are Bézout coefficients useful here?
They prove the generator is not just theoretical. The coefficients explicitly show how the generator can be built from the supplied values, which is valuable for coursework, proofs, and stepwise algebra verification.
8) Can I use negative numbers in the generator list?
Yes. The calculator accepts negative integers and normalizes the principal generator to a nonnegative value. The ideal itself does not change when a generator is multiplied by −1, so the normalized result stays mathematically consistent.