Example 1

Let A be a commutative ring and $t$ an element of A which is not a divisor of 0. Let us give A the (t)-*adic filtration* (no. 1, Example 3). Then the associated graded ring $\mathrm{gr}(A)$ is canonically isomorphic to the polynomial ring $(A/(t))[X]$. For $\mathrm{gr}_n(A) = 0$ for $n < 0$ and by definition the ring $\mathrm{gr}_0(A)$ is the ring $A/(t)$. We now note that by virtue of the hypothesis on $t$ the relation $at^n \equiv 0$ (mod. $t^{n+1}$) is equivalent to $a \equiv 0$ (mod. $t$); if $\tau$ is the canonical image of $t$ in $\mathrm{gr}_1(A)$, every element of $\mathrm{gr}_n(A)$ may then be written uniquely in the form $\alpha \tau^n$ where $\alpha \in \mathrm{gr}_0(A)$; whence our assertion.