Exercises: Factorization, reduction and decomposition problems
نویسنده
چکیده
Exercise 1 Let A be a domain, α an injective endomorphism of A, D = A[∂;α, β] a skew polynomial ring, E ∈ An×n and F ∈ An×n, R = ∂ In − E ∈ Dn×n and R′ = ∂ In − F two matrices with entries in D, M = D1×n/(D1×nR) and M ′ = D1×n/(D1×nR′) two left D-modules respectively finitely presented by R and R′. 1. Describe the left D-modules M and M ′ in terms of generators and relations. 2. Prove that any f ∈ homD(M,M ′) can be defined by means of a matrix P ∈ An×n satisfying the relation RP = QR′. 3. Deduce that Q ∈ An×n and prove that RP = QR′ is then equivalent to: { Q = α(P ), β(P ) = E P − α(P )F. (1)
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