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- Thread starter jakelyon
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- #2

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What is the cardinality of Z/2Z? Then look at the size of Z[x]/(2x).

- #3

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(2x) is the ideal consisting of all linear combinations of 2x (with integer coefficients). Now, by moding Z[x] out by (2x) it is "like" sending x to 0. So, if I am correct, then Z[x]/(2x) = Z[0] =

Z, not Z/2Z, right?

Does this make sense? How would I get Z/2Z then? Thanks.

- #4

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Maybe part of your trouble is thinking that all rings are integral domains (integral domains are rings such that if ab = 0 then a = 0 or b = 0). This ring is not and there are more familiar ones that are not either such as the ring of all matrices.

If you want to get Z/2Z from a quotient of Z[x] you would have to quotient out by the ideal (2,x). Note that Z[x] is not a principal ideal domain and this ideal cannot be generated by a single element.

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