= Square-zero extension of an algebra
= Square-zero extensions of an algebra
{synonym}
For a unital <associative algebra> $A$ and prescribed $A$-<bimodule> $M$, an extension is $0\to M\to E\to A\to0$ where $E$ is unital, $E\to A$ is unital, $M$ is a <square-zero ideal>, and the induced actions are the prescribed ones. Equivalences induce the identity on $A$ and $M$. A unital linear section produces a normalized <Hochschild cocycle> $\mu(a,b)=s(a)s(b)-s(ab)$. Conversely, $(a,m)(b,n)=(ab,an+mb+\mu(a,b))$ on $A\oplus M$ defines the extension. Changing section changes $\mu$ by a coboundary, giving classification by $HH^2(A,M)$.
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