Applied Algebra, Algebraic Algorithms and Error-Correcting by E. F. Assmus Jr. (auth.), Teo Mora (eds.)

By E. F. Assmus Jr. (auth.), Teo Mora (eds.)

In 1988, for the 1st time, the 2 foreign meetings AAECC-6 and ISSAC'88 (International Symposium on Symbolic and Algebraic Computation, see Lecture Notes in machine technological know-how 358) have taken position as a Joint convention in Rome, July 4-8, 1988. the themes of the 2 meetings are actually largely relating to one another and the Joint convention offered an outstanding celebration for the 2 learn groups to fulfill and proportion clinical reports and effects. The court cases of the AAECC-6 are incorporated during this quantity. the most issues are: utilized Algebra, idea and alertness of Error-Correcting Codes, Cryptography, Complexity, Algebra dependent equipment and purposes in Symbolic Computing and desktop Algebra, and Algebraic equipment and functions for complicated details Processing. Twelve invited papers on matters of universal curiosity for the 2 meetings are divided among this quantity and the succeeding Lecture Notes quantity dedicated to ISSACC'88. The court cases of the fifth convention are released as Vol. 356 of the Lecture Notes in desktop Science.

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Additional resources for Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 6th International Conference, AAECC-6 Rome, Italy, July 4–8, 1988 Proceedings

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Hom-Tensor Adjointness Next up: Hom-tensor adjointness and an alternative proof of right-exactness of tensor products. 1. Let R be a commutative ring, and let M , N and P be Rmodules. There are R-module isomorphisms / Ψ HomR (N, HomR (M, P )) o [α : N → HomR (M, P )] 1 N → HomR (M, P ) n → [m → β(m ⊗ n)] HomR (M ⊗R N, P ) Φ M ⊗R N → P m ⊗ n → α(n)(m) / o 1 [β : M ⊗ N → P ]. R Proof. It is straightforward to show that the map Φ is well-defined. Use the universal property for M ⊗R N to show that Ψ is well-defined.

If M is an R-module, the operators M ⊗R − and HomR (M, −) are covariant functors R M → R M. If ϕ : R → S is a homomorphism of commutative rings, then the operators S ⊗R − and HomR (S, −) are covariant functors R M → S M. 7. Let R and S be commutative rings, and let F : R M → S M be a covariant functor. φ ψ (a) F is left-exact if, for every exact sequence 0 → M − → N − → P of R-module F (φ) F (ψ) homomorphisms, the resulting sequence 0 → F (M ) −−−→ F (N ) −−−→ F (P ) of S-module homomorphisms is exact; φ ψ (b) F is right-exact if, for every exact sequence M − →N − → P → 0 of R-module F (φ) F (ψ) homomorphisms, the resulting sequence F (M ) −−−→ F (N ) −−−→ F (P ) → 0 of S-module homomorphisms is exact; φ ψ (c) F is exact if, for every exact sequence M − →N − → P of R-module homomorF (φ) F (ψ) phisms, the resulting sequence F (M ) −−−→ F (N ) −−−→ F (P ) of S-module homomorphisms is exact.

11. Let ϕ : R → S be a homomorphism of commutative rings. Prove that, if M is a finitely generated R-module, then S ⊗R M is finitely generated as an S-module. 12. Let ϕ : R → S be a homomorphism of commutative rings. Let M be an R-module, and let N be an S-module. (a) Prove that the tensor product N ⊗R M has a well-defined S-module structure given by s( i ni ⊗ mi ) = i (sni ) ⊗ mi . (b) Prove that this S-module structure is compatible with the R-module structure on N ⊗R M via restriction of scalars: for all r ∈ R and all n ∈ N and all m ∈ M , we have r(n ⊗ m) = ϕ(r)(n ⊗ m).

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