  The polynomial Szemer\'{e}di theorem implies that, for any $\delta \in
(0,1)$, any family $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y]$ of nonconstant
polynomials with constant term zero, and any sufficiently large $N$, every
subset of $\{1,\ldots, N\}$ of cardinality at least $\delta N$ contains a
nontrivial configuration of the form $\{x,x+P_1(y),\ldots, x+P_m(y)\}$. When
the polynomials are assumed independent, one can expect a sharper result to
hold over finite fields, special cases of which were proven recently,
culminating with arXiv:1802.02200, which deals with the general case of
independent polynomials. One goal of this article is to explain these theorems
as the result of joint ergodicity in the presence of asymptotic total
ergodicity. Guided by this concept, we establish, over general finite
commutative rings, a version of the polynomial Szemer\'{e}di theorem for
independent polynomials $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y_1,\ldots,
y_n]$, deriving new combinatorial consequences, such as the following. Let
$\mathcal R$ be a collection of finite commutative rings subject to a mild
condition on their torsion. There exists $\gamma \in (0,1)$ such that, for
every $R \in \mathcal R$, every subset $A \subset R$ of cardinality at least
$|R|^{1-\gamma}$ contains a nontrivial configuration $\{x,x+P_1(y),\ldots,
x+P_m(y)\}$ for some $(x,y) \in R \times R^n$, and, moreover, for any subsets
$A_0,\ldots, A_m \subset R$ such that $|A_0|\cdots |A_m| \geq
|R|^{(m+1)(1-\gamma)}$, there is a nontrivial configuration $(x, x+P_1(y),
\ldots, x+P_m(y)) \in A_0\times \cdots \times A_m$. The fact that general rings
have zero divisors is the source of many obstacles, which we overcome; for
example, by studying character sums, we develop a bound on the number of roots
of an integer polynomial over a general finite commutative ring, a result which
is of independent interest.
