Let $A$ be an algebra, $B \subset A$ a subalgebra, and $(\Omega^1(A),d)$ a first order differential calculus over $A$. Now, as is well known, the subalgebra of $\Omega^1(A)$ consisting of elements of the form $\sum_i b_{i_1}db_{i_2}$ is a differential calculus over $B$  the restriction of $\Omega^1(A)$ to $B$. My question is: For an ideal $I \subset A$, is there a suitable notion of "restriction" of $\Omega^1(A)$ to $A/I$? The obvious object to take would be $\Omega^1(A)/dI$, but it is not obvious that this object is always nontrivial.
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