$R$ is a ring and $R[x]$ is the polynomial ring over $R$ . $c$ is any fixed element of $R$ . Then the map $f(x)\mapsto f(x+c)$ is an isomorphism from $R[x]$ to itself. Now with the assumption that $R$ is a field it is written that $p(x)$ is irreducible in $R[x]$ iff $p(x+c)$ is irreducible in $R[x]$.
Now here is my question: Why do we need the assumption that $R$ is a field $?$ Since $p(x)\mapsto p(x+c)$ is already an isomorphism (irrespective of whether $R$ is a field or not) then does not irreducibility of one automatically imply irreducibility of the other $?$ What am I missing here?