By Hirosi Nagao, Yukio Tsushima

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Iii) Let εΐ5 ε2 be as above. Then the following holds: ε-V~ ε2 V (A-isomorphic) οε1Ε ~ ε2Ε (E-isomorphic). Proof. 1. (ii) If oeE, we have an A-homomorphism φ(ε2σεί): ^V-* E2V{E1V -+ε2σε1ν). Thus we have a homomorphism φ:ε2Εει^> 25 5. Endomorphism Rings H o m ^ ^ K , ε 2 Κ), which is clearly a monomorphism. , σ(ι;) = σ ^ ε ^ ) . 4, εχΕ ~ ε2Ε if and only if there exist σ,τβΕ such that (ε 2 σε 1 )(ε 1 τε 2 ) = ε 2 , (ε 1 τε 2 )(ε 2 σε 1 ) = εν But the latter condition asserts that φ(ε2σε1): εχν-^ ε2ν is an isomorphism having φ(ε 1 τε 2 ): ε2ν~+ειν as its inverse.

VA and AW, then there is a unique isomorphism g: T ^Τ' such that g o φ = φ'. Proof. By the universality property of (T, φ\ there exists a homomorphism g: T-> T such that g o φ = φ'. vxw φ ~τ VXW/ / g'°g T Likewise there exists g'\T -► T such that g' ° φ' = φ. Then it follows that (g'o g) o φ = g' o φ' = φ. But since i d r o φ = φ, we have g' ° g = i d r by the uniqueness condition. Similarly, we have g o g' = idT, and thus g is an isomorphism. ■ The above two theorems assert that the tensor product V®AW is characterized by the universality property.

Let V be a completely reducible A-module and let iel ΛεΛ, be an irreducible decomposition of V such that ViX and Vkfl are isomorphic to each other if and only if i = k. Set l/f = 0 A e A . Vik and let Vx be any one of {Va; λ e AJ, then the following holds. (i) K = 0 i e / l / , , (ii) Any irreducible A-submodule W of V is isomorphic to some Vi9 in which case it holds that W a [/,·. Consequently each Ut is the sum of all irreducible A-submodules of V isomorphic to V{. (iii) End^iOl/, = Ut. ,m} is a finite set and |A f | < oo for all i, then EndA(K) ~ End^L^) © · · · © EndA(Um) (ring isomorphism), and each EndA(Ui) is the full matrix ring of degree |A f | over the division ring End^(K).