It is a generalization to associative algebras of the notion of a separable field extension.
A homomorphism of (unital, but not necessarily commutative) rings is called separable if the multiplication map admits a section that is a homomorphism of A-A-bimodules.
It is useful to describe separability in terms of the element The reason is that a section σ is determined by this element.
The condition that σ is a section of μ is equivalent to and the condition that σ is a homomorphism of A-A-bimodules is equivalent to the following requirement for any a in A: Such an element p is called a separability idempotent, since regarded as an element of the algebra
In particular, this shows that separability idempotents need not be unique.
A field extension L/K of finite degree is a separable extension if and only if L is separable as an associative K-algebra.
The tensorands are dual bases for the trace map: if
are the distinct K-monomorphisms of L into an algebraic closure of K, the trace mapping Tr of L into K is defined by
The trace map and its dual bases make explicit L as a Frobenius algebra over K. More generally, separable algebras over a field K can be classified as follows: they are the same as finite products of matrix algebras over finite-dimensional division algebras whose centers are finite-dimensional separable field extensions of the field K. In particular: Every separable algebra is itself finite-dimensional.
If K is a perfect field – for example a field of characteristic zero, or a finite field, or an algebraically closed field – then every extension of K is separable, so that separable K-algebras are finite products of matrix algebras over finite-dimensional division algebras over field K. In other words, if K is a perfect field, there is no difference between a separable algebra over K and a finite-dimensional semisimple algebra over K. It can be shown by a generalized theorem of Maschke that an associative K-algebra A is separable if for every field extension
If K is commutative ring and G is a finite group such that the order of G is invertible in K, then the group algebra K[G] is a separable K-algebra.
There are several equivalent definitions of separable algebras.
A K-algebra A is separable if and only if it is projective when considered as a left module of
[2] Moreover, an algebra A is separable if and only if it is flat when considered as a right module of
Separable algebras can also be characterized by means of split extensions: A is separable over K if and only if all short exact sequences of A-A-bimodules that are split as A-K-bimodules also split as A-A-bimodules.
arising in the definition above is a A-A-bimodule epimorphism, which is split as an A-K-bimodule map by the right inverse mapping
The converse can be proven by a judicious use of the separability idempotent (similarly to the proof of Maschke's theorem, applying its components within and without the splitting maps).
[3] Equivalently, the relative Hochschild cohomology groups
Any ring R with elements a and b satisfying ab = 1, but ba different from 1, is a separable extension over the subring S generated by 1 and bRa.
If K is commutative, A is a finitely generated projective separable K-module, then A is a symmetric Frobenius algebra.
[4] Any separable extension A / K of commutative rings is formally unramified.
The converse holds if A is a finitely generated K-algebra.
[5] A separable flat (commutative) K-algebra A is formally étale.
[6] A theorem in the area is that of J. Cuadra that a separable Hopf–Galois extension R | S has finitely generated natural S-module R. A fundamental fact about a separable extension R | S is that it is left or right semisimple extension: a short exact sequence of left or right R-modules that is split as S-modules, is split as R-modules.
Usually relative properties of subrings or ring extensions, such as the notion of separable extension, serve to promote theorems that say that the over-ring shares a property of the subring.
For example, a separable extension R of a semisimple algebra S has R semisimple, which follows from the preceding discussion.
There is the celebrated Jans theorem that a finite group algebra A over a field of characteristic p is of finite representation type if and only if its Sylow p-subgroup is cyclic: the clearest proof is to note this fact for p-groups, then note that the group algebra is a separable extension of its Sylow p-subgroup algebra B as the index is coprime to the characteristic.
The separability condition above will imply every finitely generated A-module M is isomorphic to a direct summand in its restricted, induced module.
But if B has finite representation type, the restricted module is uniquely a direct sum of multiples of finitely many indecomposables, which induce to a finite number of constituent indecomposable modules of which M is a direct sum.
The converse is proven by a similar argument noting that every subgroup algebra B is a B-bimodule direct summand of a group algebra A.