How does one prove that the real radical of an ideal $I$ in the real polynomial ring $\mathbb{R}[x_1,\ldots,x_n]$ is an ideal. (NOT to be confused the radical of an ideal)
I am using the wikipedia definition of the real ideal as well.
Consider $f,h$ in the real radical. both have a corresponding integers $m,n$ and a sequence of functions such that its sum of squares are in the ideal $I$. in other words, $f^{2m},g^{2n}$ can be written as a sum of squared functions plus a function in ideal I. consider the expansion of $(f+h)^{mn}$. the goal is to show that this expansion can be written as a sum of squared functions plus a sum of functions in ideal $I$. But when you expand it, this is not at all clear. even if you consider $(f+h)^{2m+2n}$, its not clear either.
Is there are more intuitive approach to showing this?