Dominio euclideo

In algebra, un dominio euclideo o anello euclideo è un anello commutativo su cui è possibile effettuare una divisione euclidea.

Definizione

Un dominio d'integrità è un anello euclideo se è possibile definire una funzione che associa ad ogni elemento non nullo un numero naturale con le seguenti proprietà:

  • ,con tali che e oppure

In pratica, tale proprietà dice che è sempre possibile effettuare una divisione fra due numeri non nulli a e b, avente quoziente q e resto r, tale che il resto r sia "più piccolo" di b: esattamente quanto accade con i numeri interi. L'essere "più piccolo" è realizzato dalla funzione , detta valutazione o norma.

Esempi

Anelli euclidei

  • l'anello Z degli interi, con v(n) = |n| il valore assoluto di n;
  • l'anello Z[i] degli interi gaussiani, con v(a+bi) = a2+b2;
  • l'anello K[X] dei polinomi a coefficienti in un campo K, con v(p) uguale al grado del polinomio p;
  • l'anello K[[X]] delle serie formali di potenze a coefficienti in un campo K, con v(f) uguale al grado del più piccolo monomio presente nella serie f.
  • un campo qualsiasi, semplicemente con v(x) = 1 per ogni x.

Anelli non euclidei

  • un anello che non è ad ideali principali non è neppure euclideo. Più difficile è trovare un anello ad ideali principali che non sia euclideo: un esempio è dato da

Proprietà

Sia A un anello euclideo.

  • A è un anello ad ideali principali. Infatti ogni ideale I è generato da uno qualsiasi degli elementi in I avente valutazione minima;
  • l'algoritmo di Euclide per trovare il massimo comune divisore fra due elementi funziona in A;
  • è possibile trovare una valutazione su A tale che v(ab) ≥ v(a) per ogni a e b non nulli;
  • poiché A è ad ideali principali, è anche un anello a fattorizzazione unica: una valutazione con la proprietà v(ab) ≥ v(a) può essere usata per trovare direttamente la fattorizzazione.

Bibliografia

  • Giulia Maria Piacentini Cattaneo, Algebra - un approccio algoritmico. Decibel-Zanichelli, Padova 1996, ISBN 978-88-08-16270-0
  • Luca Barbieri Viale, §5.4 Anelli euclidei, Che cos'è un numero ? Raffaello Cortina, 2013, ISBN 978-88-6030-604-3

Voci correlate

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