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# If <math>x \in (r)</math> and <math>y \in (r)</math>, then we can write <math>x = ra</math> and <math>y = rb</math>. Then <math>x+y = ra + rb = r(a+b) \in R</math>.
# If <math>x \in (r)</math> and <math>y \in (r)</math>, then we can write <math>x = ra</math> and <math>y = rb</math>. Then <math>x+y = ra + rb = r(a+b) \in R</math>.
# If <math>x \in (r)</math> and <math>b \in R</math> is an arbitrary element of the ring <math>R</math>, then <math>xb = (ar)b = (ab)r \in R</math>.
# If <math>x \in (r)</math> and <math>b \in R</math> is an arbitrary element of the ring <math>R</math>, then <math>xb = (ar)b = (ab)r \in R</math>.
When <math>R=\mathbb Z</math> is the ring of integers, then every ideal is a principal ideal: that is, any ideal <math>I</math> of the ring of integers is the set of elements "multiples of k" for some integer k. Many familiar rings have the property that every ideal is a principal ideal: such rings are known as '''principal ideal domains'''.
[[Category:Mathematics]]
[[Category:Mathematics]]