Polynomials over a ring evaluated at a value?

[Bipolarity]

In ring theory, a polynomial over a rings, say $R[x]$ is presented as an abstract object of the form:
$p(x) = a_{n}x^{n} + ...+ a_{1}x + a_{0}$ where the coefficients $a_{n}...a_{0}$ are from a ring R with unity and $x$ is a formal symbol.

So what is the significance of $p(x+1)$ ? In a high school algebra, one would simply interpret this as $p(x)$ with every instance of the variable $x$ replaced by $x+1$. But in this notation, what does $x + 1$ even mean? It is itself a one-degree polynomial of $R[x]$ but then what does the notation $p(x+1)$ refer to?

BiP

 

[HallsofIvy]

It means exactly what it appears to mean. Since R is a ring with unity, "1" is that unity (multiplicative identity) and x+ 1 means the sum of some member, x, of the ring with that multiplicative identity. Since a ring is "closed under addition" x+ 1 is again a member of the ring and, since all the coefficients are members of the ring, all multiplications and addition in "p(x+ 1)" are defined in the ring.