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Proposition 3.2.16.
Let K be a field and A be an associative unital K-algebra. Then the following properties are
equivalent:
1. A is projective as an A e -module.
2. The short exact sequence of A e -modules
0 → ker μ → A e μ → A → 0
splits. Put differently, the multiplication epimorphism
μ : A ⊗ K A → A
has a right inverse as a morphism of bimodules:
ϕ : A → A ⊗ K A
with μ ◦ ϕ = id A and ϕ(abc) = a ∙ ϕ(b) ∙ c for all a, b, c ∈ A.
3. Given any extension of fields K ⊂ E, the induced E-algebra A ⊗ K E is semisimple.
For the proof of this statement, we refer to Chapter 10 of R.S. Pierce, Associative Algebras,
Springer Graduate Texts in Mathematics 88.
Definition 3.2.17
A K-algebra A that has one of the properties of proposition 3.2.16 is called separable.
Remarks 3.2.18.
1. The choice of a right inverse ϕ of μ is called the choice of a separability structure of A.
2. We can describe ϕ by the element e := ϕ(1) ∈ A e . Indeed, since ϕ is a morphism of
A-bimodules, ϕ(a) = (a ⊗ 1)e. Obviously, μ(e) = 1 and (a ⊗ 1)e = e(1 ⊗ a) for all a ∈ A.
With the multiplication in A e , we have e 2 = e. The element e ∈ A e is therefore called a
separability idempotent.
3. Separable algebras over fields are finite-dimensional and semisimple.
More precisely, a K-algebra A is separable, if and only if
A ∼ = A 1 ⊕ ∙ ∙ ∙ ⊕ A r
is a direct sum of finite-dimensional simple K-algebras where all Z(A i )/K are separable
extensions of fields.
4. We present an example: for any field K, the full matrix algebra Mat n (K) is a separable
K-algebra.
Introduce matrix units ɛ ij , i.e. ɛ ij is the matrix with zero entries, except for one in the
i-th line and j-th column. Fix some index 1 ≤ j ≤ n; then
e (j) :=
n∑
ɛ ij ⊗ ɛ ji ∈ Mat n (K) opp ⊗ Mat n (K)
i=1
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