$\mathbb{Z} [x] $as union of three proper subring.

Question

We know that the union of two subrings is a subring if and only if either of the subring is contained in the other. so there's no Ring $R$ as union of two proper subring. Now let R be finte ring then we can write R as union of three proper subring if $|R|=4,8,16$. For infinite ring( for example $\mathbb{Z}[x]$ ) Can we write $\mathbb{Z} [x] $as union of three proper subring ?

Answer 1

I assume that we are talking about unital subrings.

In fact, $\mathbb Z [X]$ cannot be a union of any number of proper subrings.

Suppose $\mathbb Z [X] = \bigcup R_i$, all $R_i$ being proper subrings. Then, $X \in \bigcup R_i$ means that $X \in R_i$ for at least one $i$. But $X$ generates the whole $\mathbb Z [X]$ as a ring, thus $\mathbb Z [X] \subset R_i$, which contradicts $R_i$ being a proper subring.