Source
Wedderburn (1908), “On hypercomplex numbers”, Proceedings of the London Mathematical Society (2) 6:77–118, LIT-309, read in full in NOTE-280.
What was actually shown
The setting. Associative algebras with a finite basis over a field F that is “constant throughout but otherwise arbitrary” (§1). Semi-simple means no nilpotent invariant sub-algebra, that is no nilpotent two-sided ideal (§4).
The radical (Thm 13). A maximal nilpotent ideal N contains all others, so A/N is semi-simple.
Into simple blocks (Thms 10, 16, 17). A semi-simple algebra has an identity and, if not simple, splits along central idempotents. It is therefore the direct sum A₁ + ⋯ + A_n of simple algebras with A_pA_q = 0 for p ≠ q, uniquely.
Each block is a matrix algebra (Thms 20–23). In a simple algebra the primitive idempotents e_p give Peirce pieces e_pAe_q with A_pqA_qr = A_pr (Thm 20). Choosing x_pq with x_pqx_qp = e_p gives n² matrix units e_pqe_qr = e_pr (Thm 21), and A is the direct product of the corner algebra A₁₁ with the full matrix algebra (e_pq) (Thm 22). A₁₁ is primitive: it has one idempotent, so every element is invertible or nilpotent (Thm 15), and being semi-simple it is a division algebra. Conversely such a product is simple, and only multiples of the identity commute with a full matrix algebra (Thm 23 and corollary).
Idempotents without field extension (Thm 14). Every algebra that is not nilpotent contains an idempotent rational in F. This is what replaces the characteristic roots of Cartan’s and Frobenius’s methods, which needed ℂ or ℝ.
What could have come out otherwise. Over a field that is not algebraically closed the decomposition could have failed to exist or to be unique. It does neither.
What this does not say
- That every block is a matrix algebra over the field. The division factor can be non-trivial. Over ℝ it can be ℂ or ℍ, and the paper’s own §11 has a quaternion example. A real *-algebra’s block can be M_n(ℂ) or M_n(ℍ), which LIT-352’s Theorem 3.1(C) forgets (NOTE-280).
- That the paper speaks of a centre or of isotypic components. It works with algebras, not modules, and never uses the word “centre”. That Z(A) is spanned by the block identities follows from Thm 23’s corollary and the uniqueness of the blocks; “types = Z(𝒜)” is a modern gloss.
- That every algebra is nilpotent plus semi-simple. Summary (iii) claims it, citing Thms 24 and 28, which prove it only when A/N is simple or primitive. Thm 28’s proof needed a correction added on 1 February 1908.
- Anything beyond a finite basis. The generalisation to rings with chain conditions is Artin’s (LIT-331).
- A classification of division algebras. The paper concedes it leaves them unclassified (§6).
Connections
- THEORY-019 needs the commutant of an isotypic component to be M_m(𝔽) with
𝔽 = ℝ, ℂ or ℍ, and its
promote_whenasks for a held source that proves it. This theorem supplies the algebra-side half: every simple real block is a full matrix algebra over a real division algebra. That ℝ, ℂ and ℍ are the only real division algebras is Frobenius’s theorem, which neither the paper nor the record holds. It does not supply the representation-side step, that the commutant of a representation is such a block. - LIT-313’s Theorem 3, on simple weakly closed rings being factors, is the infinite-dimensional descendant of the trivial-centre corollary (THEORY-592).