Is the ring of formal power series in infinitely many variables a unique factorization domain?

Is $R[[x_1,x_2\dots]]$ a unique factorization domain where the notation means infinite sums where each term is a finite product over the $x_i's$ with coefficients in $R$.

I am most interested in the case where $R = \Bbb R$ or $\Bbb C$.

Solution 1:

The answer is yes, for sufficiently nice $R$ (including the case where $R$ is a field). This was proved by Hajime Nishimura in the paper "On the unique factorization theorem for formal power series", J. Math. Kyoto Univ. Volume 7, Number 2 (1967), 151-160, available in open access on Project Euclid.

The precise statement for countably many variables is as follows:

If the ring $R[[x_1, \ldots, x_n]]$ is a unique factorization domain for any positive integer $n$, then so is $R[[x_1, x_2, \ldots]]$.

The ring $R[[x_1, x_2, \ldots]]$ is denoted by $R\{X\}_{\aleph_0}$ in the paper, where the above is found as (a special case of) Theorem 1.

In particular, the result is true if $R$ is a field, or even a principal ideal domain.