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text: '[summary: A principal ideal domain is an [-5md] in which every [ideal_ring_theory ideal] has a single generator.]\n\nIn [3gq ring theory], an [-5md] is a **principal ideal domain** (or **PID**) if every [ideal_ring_theory ideal] can be generated by a single element.\nThat is, for every ideal $I$ there is an element $i \\in I$ such that $\\langle i \\rangle = I$; equivalently, every element of $I$ is a multiple of $i$.\n\nSince ideals are [5r6 kernels] of [ring_homomorphism ring homomorphisms] ([5r9 proof]), this is saying that a PID $R$ has the special property that *every* ring homomorphism from $R$ acts "nearly non-trivially", in that the collection of things it sends to the identity is just "one particular element, and everything that is forced by that, but nothing else".\n\n# Examples\n\n- Every [euclidean_domain Euclidean domain] is a PID. ([euclidean_domain_is_pid Proof.])\n- Therefore $\\mathbb{Z}$ is a PID, because it is a [euclidean_domain Euclidean domain]. (Its Euclidean function is "take the modulus".)\n- Every [481 field] is a PID because every ideal is either the singleton $\\{ 0 \\}$ (i.e. generated by $0$) or else is the entire ring (i.e. generated by $1$).\n- The [polynomial_ring ring $F[X]$ of polynomials] over a field $F$ is a PID, because it is a Euclidean domain. (Its Euclidean function is "take the [polynomial_degree degree] of the polynomial".)\n- The ring of [gaussian_integer Gaussian integers], $\\mathbb{Z}[i]$, is a PID because it is a Euclidean domain. ([gaussian_integers_is_pid Proof]; its Euclidean function is "take the [norm_complex_number norm]".)\n- The ring $\\mathbb{Z}[X]$ (of integer-coefficient polynomials) is *not* a PID, because the ideal $\\langle 2, X \\rangle$ is not principal. This is an example of a [-unique_factorisation_domain] which is not a PID. [todo: proof of this]\n- The ring $\\mathbb{Z}_6$ is *not* a PID, because it is not an integral domain. (Indeed, $3 \\times 2 = 0$ in this ring.)\n\nThere are examples of PIDs which are not Euclidean domains, but they are mostly uninteresting.\nOne such ring is $\\mathbb{Z}[\\frac{1}{2} (1+\\sqrt{-19})]$. ([Proof.](http://www.maths.qmul.ac.uk/~raw/MTH5100/PIDnotED.pdf))\n\n# Properties\n\n- Every PID is a [-unique_factorisation_domain]. ([principal_ideal_domain_has_unique_factorisation Proof]; this fact is not trivial.) The converse is false; see the case $\\mathbb{Z}[X]$ above.\n- In a PID, "[5m2 prime]" and "[5m1 irreducible]" coincide. ([5mf Proof.]) This fact also characterises the [maximal_ideal maximal ideals] of PIDs.\n- Every PID is trivially [noetherian_ring Noetherian]: every ideal is not just *finitely* generated, but generated by a single element.',
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