# Extension of an associative algebra

$R$ over a commutative ring $K$

A homomorphism $\phi : S \rightarrow R$ of a $K$- algebra $S$ onto $R$. If $\mathop{\rm Ker} \phi = I$ is an algebra with zero multiplication, then the extension is called singular. In this case, $I$ is an $R$- module in a natural way. The set of all extensions of $R$ with kernel $I$ admits an equivalence relation (the same as for groups, modules, etc.), and the set of equivalence classes of extensions is denoted by $F ( R, I)$. If the algebra $R$ is $K$- projective, then the algebra $S$ splits into a direct sum of $K$- modules, $S = I + R$, and the elements of $S$ can be written as pairs $( u , r)$, $u \in I$, $r \in R$, with multiplication given by

$$( u _ {1} , r _ {1} ) ( u _ {2} , r _ {2} ) = \ ( u _ {1} r _ {2} + r _ {1} u _ {2} + a ( r _ {1} , r _ {2} ),\ r _ {1} r _ {2} ),$$

where $a: R \otimes R \rightarrow I$. The associativity of multiplication imposes restrictions on $a$ that make it into a cocycle. Mapping an extension to its cocycle defines an isomorphism of $K$- modules between $F ( R, I)$ and the second cohomology group $H ^ {2} ( R, I)$ of $R$ with coefficients in $I$.

In a completely different context, any algebra containing $R$ is also called an extension of $R$. Such extensions are often connected with a specific construction (polynomials over $R$, localization of $R$, ring of partial fractions of the algebra $R$, etc.). See also Extension of a field.

#### References

 [1] S. MacLane, "Homology" , Springer (1963) [2] G. Hochschild, "On the cohomology groups of an associative algebra" Ann. of Math. , 46 (1945) pp. 58–67

The cohomology group $H ^ {2} ( R , I )$ is also known as the Hochschild cohomology (group) of $R$ with values in $I$.