Irreducible element

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In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.

Irreducible elements should not be confused with prime elements. (A non-zero non-unit element a in a commutative ring R is called prime if, whenever a | bc for some b and c in R, then a|b or a|c.) In an integral domain, every prime element is irreducible,[1][2] but the converse is not true in general. The converse is true for UFDs[2] (or, more generally, GCD domains.)

Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal. However, if D is a GCD domain, and x is an irreducible element of D, then the ideal generated by x is an irreducible ideal of D.[3]

Example[edit]

In the quadratic integer ring \mathbf{Z}[\sqrt{-5}], it can be shown using norm arguments that the number 3 is irreducible. However, it is not a prime element in this ring since, for example,

3 | \left(2 + \sqrt{-5}\right)\left(2 - \sqrt{-5}\right)=9

but 3 does not divide either of the two factors.[4]

References[edit]

  1. ^ Consider p a prime that is reducible: p=ab. Then p | ab => p | a or p | b. Say p | a => a = pc, then we have: p=ab=pcb => p(1-cb)=0. Because R is an integral domain we have: cb=1. So b is a unit and p is irreducible.
  2. ^ a b Sharpe (1987) p.54
  3. ^ http://planetmath.org/encyclopedia/IrreducibleIdeal.html
  4. ^ William W. Adams and Larry Joel Goldstein (1976), Introduction to Number Theory, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9