Subrings of the formal power series

Question

I was struggling with subrings, and I was wondering if anyone could clarify this problem for me

$S=\{a_0 + a_2x^2 + a_4x^4+\dotsm\mid a_i \in\mathbb Q\}\subset \mathbb Q[[x]]$

where $\mathbb Q$ is the field of rational numbers.

So my lecture said it is a subring iff
for all $a,b$ element of $S$ we have $a-b$ is an element of $S$, for all $a,b$ element of $S$ we have $a b$ is an element of $S$ .

So I believe this is not a subring because $a_0 - a_2x^2$ is not an element of $S$ but I don't know if I'm 100% right or even how to prove this, can someone please explain?

Answer 1

Your set $\;S\;$ contains all the power series for which all the coefficients of odd powers of $\;x\;$ are zero...or in other, simpler perhaps, words: all the power series with only even powers (meaning: only the coefficients of even powers are, perhaps, non-zero).

1) Is the difference of two such series as above described again an element in $\;S\;$ ?

2) Is the product of two such elements as above described again an element of $\;S\;$ .

The answer for both question is yes ...but can you prove this ?

Answer 2

Hint for the product:

The product for two formal powers series is defined as a Cauchy product.

The coefficient of $x^n$ in the product is $$ \sum_{i+j=n} a_i\mkern1mub_j.$$ If $i+j=n$ and $n$ is odd, can both $i$ and $j$ be even?