Let $K$ be a commutative but not necessarily unital ring. Let $C$ be a unital commutative ring.

An extension is the data of a unital commutative ring $R$, a homomorphism $\phi:K\rightarrow R$, a unital homomorphism $\psi:R\rightarrow C$ such that $\mathrm{Im}\:\phi=\mathrm{Ker}\:\psi$. Two extensions are said to be isomorphic if there is an isomorphism $R_1\approx R_2$ making the rhomb-like diagram commute.

Is there a standard name for the algebraic gadget parametrizing the isomorphism classes of extensions? Is there a textbook or an article where I can read about it?