# Extensions of rings

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?

• Have a look at EGA IV.1, §18 (of Chapter 0): "Compléments sur les extensions d'algèbres". – abx Jun 28 at 10:13