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## Main Question or Discussion Point

Can anyone explain, in detail, why/why not Z[X]/(2x) is isomorphic to Z/2Z? I know that every element in Z[x] can be written as a_0 + a_1 x + a_2 x^2 + ... with a_i in Z and only finitely many a_i's are nonzero. Now, does (2x) = (2, 2x, 2x^2,...)? Also, the quotient is "like" taking 2x=0, or x=0. Thus, I think that all elements of Z[x]/(2x) would look like a_0/2 for some a_0 in Z. But this does not give Z/2Z does it? Thanks.