Principle Ideal Ring
Theorem:- If {(Ii,+,.)} is the collection of ideals of the ring (R,+,.).Then (?Ii,+,.) is also an ideal of R .
Proof:-since ?i 0?Ii ,so 0??Ii ,hence ?Ii??.
Let a,b??Ii , r?R.
a,b ?Ii ,?I so a-b?Ii ?i.
_? a-b??Ii.
Let a??Ii , r?R
_a?Ii , ?i ,r?R.
So a.r ?Ii , ?i
? a.r,r.a ??Ii
?(?Ii,+,.) is an ideal of (R,+,.).
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