trying to understand what a polynomial ring is I don' really understand what a polynomial ring is, maybe because the lack of examples.
Consider for example the polynomial ring $\mathbb{Z}[x]$. Can you please tell me how this polynomial ring (its elements) looks like? How is x defined, is $x \in \mathbb{Z}$? What are the two operations of the ring?
 A: If you really need a formal definition, the ring $R[x]$ of polynomials over a (commutative) ring $R$ is defined as the set of all functions $f:\mathbb N \to R$ that have finite support, that is, $f(n)=0$ for all but a finite number of $n$. (Here $\mathbb N$ contains $0$.) That set is the same as the set of all sequences in $R$ that are eventually $0$.
The zero element in $R[x]$ is $(0,0,0,\dots)$.
The identity element in $R[x]$ is $(1,0,0,\dots)$.
$x$ is defined as $(0,1,0,\dots)$.
Addition is defined componentwise and multiplication is defined by $x^n=(0,0,\dots,1,0,\dots)$, where the $1$ is at the $n$-th position, and then expanded by linearity to general polynomials.
The result is that $f=(a_0,a_1,a_2,\dots,a_n,0,\dots)$ is the same as $a_0+a_1 x +a_2 x^2 + \cdots + a_n x^n$, as expected.
If you drop the requirement of finite support, you get the ring of formal power series over $R$.
If $R$ is not commutative, you need to declare that $x$ commutes with $R$.
But you really should focus on the operational definition: $R[x]$ is a ring that contains $R$ and $x$, and $x$ is subject to no other requirements except that is commutes with $R$, and what follows from the ring axioms. This implies the universal property of $R[x]$: If $R$ is a subring of $A$ and $a\in A$ commutes with $R$, then there is a unique ring homomorphism $R[x]\to A$ that fixes $R$ and sends $x$ to $a$. We also say that $R[x]$ is the free $R$-algebra over one element.
A: If $R$ is a ring, then the polynomial ring $R[X]$ is the ring of all polynomials with coefficients in $R$
$$R[X] = \{a_0 + a_1 X + \ldots + a_n X^n : a_i \in R\ \forall i\}$$
with the usual addition and multiplication of polynomials:
$$\sum_{i=0}^na_iX^i+\sum_{i=0}^nb_iX^i=\sum_{i=0}^n(a_i+b_i)X^i \quad\text{(some of the $a_i, b_i$ may be $0$)}\\\sum_{i=0}^na_iX^i\cdot\sum_{j=0}^mb_jX^j=\sum_{i=0}^n\sum_{j=0}^ma_ib_jX^{i+j}$$
I usually use a capital $X$ rather than $x$ for elements of this ring to stress that this $X$ is a formal symbol that takes no specific value. Elements of this ring are formal polynomials, where we treat each polynomial as an object in its own right - we don't sub in any values for $X$.
A: Given a ring $K$, the polynomial ring $K[x]$ for some transcendental $x$ over $K$ is the ring of polynomials $\sum_{i=0}^n a_i x^i$ in $x$ with coefficients from $K$. 
Given two polynomial $f(x) = \sum_{i = 0}^n a_i x^i$ and $g(x) = \sum_{i = 0}^m b_i x^i$ in $K[x]$, addition and multiplication is defined by $f(x) + g(x) = \sum_{i = 0}^m (a_i + b_i) x^i$ and $f(x) \cdot g(x) = \sum_{i = 0}^n \sum_{j = 0}^m a_i b_i x^{i + j}$ where $m \geq n$, with addition and multipliation of coefficients defined by the usual ring operations of $K$.
So elements of $\Bbb Z[x]$ would look like $a_n x^n + \cdots + a_1 x + a_0$ for $a_i \in \Bbb Z$, i.e., polynomials in $x$ with integer coefficients.
A: An example element of $\Bbb Z[x]$ looks like this:
$$
x^3 - 5x^2 + 3x + 11
$$
and addition and multiplication works the way it has always done:
$$
(x^2 + 3x - 5) + (3x^2 + 7) = 4x^2 + 3x + 2
$$
and
$$
(x^2 + 3x - 5)\cdot (3x^2 + 7) = x^2(3x^2 + 7) + 3x(3x^2 + 7) -5(3x^2 + 7)\\
= 3x^4 + 7x^2 + 9x^3 + 21x -15x^2 - 35\\
= 3x^4 + 9x^3 -6x^2 + 21 x - 35
$$
The symbol $x$ is just an $x$, it's not in and of itself defined to be anything more than a symbol. It's there to help you sort through the multiplication and addition above.
