Z[x] = ring of polynomials over the integers
Define the subring, R, of Z[x] by
R = {all f(x) on Z[x] such that f(x) = a0 + a2 x^2 + . . . + an x^n, for some a0, a2, . . ., an on Z, and some even nonnegative integer, n}
We are asked to prove that the R-module Z[x] does not decompose into a direct sum of cyclic submodules. However, I do not see why this is true.
Why is it not true that
R-module Z[x] = direct sum of the cyclic submodules <1> and <x>,
where
<1> = R1 = R = {a0 + a2 x^2 + . . . + an x^n | a0, a2, . . ., an are on Z and n is an even nonnegative integer}
and
<x> = Rx = {a1 x + a3 x^3 + . . . + am x^m | a1, a3, . . ., am are on Z and m is an odd nonnegative integer}
It would seem that any element, g(x) = b0 + b1 x + . . . + bk x^k, of Z[x], where b0, b1, . . ., bk are integers, and k is a nonnegative integer, could be written as a unique sum of an element of <1> with an element of <x>
(just form an element from <1> by choosing the terms of g(x) with even powers of x, and form an element from <x> by choosing the terms of g(x) with odd powers of x).
Define the subring, R, of Z[x] by
R = {all f(x) on Z[x] such that f(x) = a0 + a2 x^2 + . . . + an x^n, for some a0, a2, . . ., an on Z, and some even nonnegative integer, n}
We are asked to prove that the R-module Z[x] does not decompose into a direct sum of cyclic submodules. However, I do not see why this is true.
Why is it not true that
R-module Z[x] = direct sum of the cyclic submodules <1> and <x>,
where
<1> = R1 = R = {a0 + a2 x^2 + . . . + an x^n | a0, a2, . . ., an are on Z and n is an even nonnegative integer}
and
<x> = Rx = {a1 x + a3 x^3 + . . . + am x^m | a1, a3, . . ., am are on Z and m is an odd nonnegative integer}
It would seem that any element, g(x) = b0 + b1 x + . . . + bk x^k, of Z[x], where b0, b1, . . ., bk are integers, and k is a nonnegative integer, could be written as a unique sum of an element of <1> with an element of <x>
(just form an element from <1> by choosing the terms of g(x) with even powers of x, and form an element from <x> by choosing the terms of g(x) with odd powers of x).