Every nonzero element in a PID A factorizes as a product of irreducible elements.
Proof: By way of contradiction, assume there exist a nonzero element b A such that it
doesnt factorizes as a product of irreducible elements. Therefore, b itself isnt irredu
Suppose
R is a PID and a1, a2, , an are non-zero elements of
R. Consider I = (
a1, a2, , an ) = cfw_ r1 * a1 + r2 * a2 +.+ rn * an : ri R, i be the ideal
generated by a1, a2, , an . Since R is a PID, then I = (d) for some d R. Show
d is the greatest co
Every nonzero element in a PID A factorizes as a product of irreducible elements.
Proof: By way of contradiction, assume there exist a nonzero element b A such that it
doesnt factorizes as a product of irreducible elements. Therefore, b itself isnt irredu
Were interested to prove that
First, we declare that I
I
J
IJ .
J IJ . Let y I J . Note that y2 = yy IJ, since y I , J .
Thus, y IJ . Therefore, we have shown that I
On the other hand, let x
I
I
J IJ .
J . Thus, there exist a natural number m such that