Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniquely separable extensions   The separability tensor element of a separable extension of noncommutative
rings is an idempotent when viewed in the correct endomorphism ring; so one
speaks of a separability idempotent, as one usually does for separable
algebras. It is proven that this idempotent is full if and only the H-depth is
1 (H-separable extension). Similarly, a split extension has a bimodule
projection; this idempotent is full if and only if the ring extension has depth
1 (centrally projective extension). Separable and split extensions have
separability idempotents and bimodule projections in 1 - 1 correspondence via
an endomorphism ring theorem in Section~3. If the separable idempotent is
unique, then the separable extension is called uniquely separable. A Frobenius
extension with invertible $E$-index is uniquely separable if the centralizer
equals the center of the over-ring. It is also shown that a uniquely separable
extension of semisimple complex algebras with invertible E-index has depth 1.
Earlier group-theoretic results are recovered and related to depth $1$. The
dual notion, uniquely split extension, only occurs trivially for finite group
algebra extensions over complex numbers.

11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
37
The categories of this paper are: supr-con