Extension of R-modules.

Question

Let $$ 0\longrightarrow{A}\longrightarrow{B}\longrightarrow{C}\longrightarrow{0} $$ be a short exact sequence of $R$-modules $A,B,C$ then why do we call $B$ is an extension of $C$ by $A$?

Answer 1

I usually say that

$B$ is an extension by $C$ of $A$

rather than conversely, because $A$ is (isomorphic to) a submodule of $B$. However the terminology can vary.

The cokernel of the extension is often mentioned first, because this recalls the order in the “Ext group”, $\operatorname{Ext}^1_R(C,A)$: any exact sequence $0\to A\to B\to C\to 0$ is a representative of an element in this (abelian) group.

Answer 2

My impression is that group theorists often say that a group is an extension of its quotients, while the rest of us say that a group is an extension of its subgroups.