$x+2$ is irreducible in the power series ring $\mathbb{Z}[[x]]$ For the last few days I am trying to prove that $x+2$ is irreducible in $\mathbb{Z}[[x]]$. I think that it is false...  I would be very much thankful for any kind of suggestions and help. 
 A: Recall that the invertible elements are those where the constant coefficient is invertible in the integers, so the constant coefficient is $\pm 1$. (Your element is thus certainly not invertible; though we do not really needs this.)
Assume a factorization $(x+2) = f g$. Let us write $f= a_0 + Xf_1$ and $g= b_0 + X g_1$ with integers $a_0,b_0$ and powerseries $f_1, g_1$. So $2 = a_0 b_0$. Thus $a_0$ or $b_0$ is $\pm 1$ and again by the above claim you now get that the element is irreducible since one of the factors is invertible. 
A: The arithmetic of power series rings $R[[X]]$ where $R$ is an integral domain is very simple. The formula $(1-aX)^{-1}=\sum_{i\in\Bbb N}a^iX^i$ (where $a$ is any power series) shows that any series with constant term$~1$, and therefore any series with invertible (in$~R$) constant term is invertible. This means that the map $R[[X]]\to R$ that selects the constant coefficient not only maps invertible elements to invertible elements (as any unitary ring morphism does) but also maps non-invertible elements to non-invertible elements. This means that any element that maps to a nonzero element $r\in R$ has the same invertible/irreducible/reducible status as $r$. You element $x+2$ maps to the prime $2$, so it is irreducible.
The only irreducibles not covered by this are those that map to$~0$. But it is not hard to see there is only one class of these namely the one of $X$: the power series that start with $X$ with an invertible coefficient.
A: Hint $ $ Let $\ h(c_0 + c_1 x+\cdots) = c_0.\ $ Apply the following consequence of unit-faithful homs
Lemma $\,\ $ If $\,h\,$ is a monoid hom such that $h(x)$ unit $\,\color{#0a0}\Rightarrow\, x\,$ unit then $\,\color{#c00}{h(a)\,\ \rm irred}$ $\,\Rightarrow\, a\,$ irred
Proof $\,\ a = bc\,\Rightarrow\, ha = hb\ hc\,\color{#c00}{\Rightarrow} hb\,$ unit or $\,hc\,$ unit $\color{#0a0}\Rightarrow\, b\,$ unit or $\,c\,$ unit $\,\Rightarrow\, a\,$ irred
