Let R = {(a,, a,, a, ...)}, where each a; E Z. Let I = {(a,, a, az, ...)}, where only a finite number of terms are nonzero. Prove that I is not a principal ideal of R.
Here given that , where each .
Consider, .
It has to be proved that is not principal ideal of .
Consider, is a principal ideal of that is an element that exists in such that
, here is defined as ideal and it is generated by the .
Since, is generated by the , then each element of must be the form for the some in
Since, contains finitely -many non-zero terms, hence also have finitely many non-zero terms.
Consider, , here is last non-zero term.
Now for any in .
Therefore, we have .
Consider the element in
As there are many infinitely non-zero terms in , cann't be in .
However, is a principal ideal, and some element must be existed in .
Since, is generated by , then for some in .
It follows, there are many infinitely non-zero terms in , it contradicts that is generated by
And each element of is the form of having many finitely non-zero terms.
Hence, it can be concluded that is not ideal principal of .
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