Projective and injective properties of representations of a quiver Q= •→•→•
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We define injective and projective representations of a quiver Q= •→•→•. Then we show that a representation [수식] of a quiver Q= •→•→• is projective if and only if each M₁, M₂, M ₃is projective left R-module and f₁(M₁) is a summand of M₂and f₂(M₂) is a summand of M₃. And we show that a representation [수식] of a quiver Q= •→•→• is injective if and only if each M₁,M₂,M₃ is injective left R-module and ker(f₁) is a summand of M₁ and ker(f₂) is a summand of M₂.
Abstract
We define injective and projective representations of a quiver Q= •→•→•. Then we show that a representation [수식] of a quiver Q= •→•→• is projective if and only if each M₁, M₂, M ₃is projective left R-module and f₁(M₁) is a summand of M₂and f₂(M₂) is a summand of M₃. And we show that a representation [수식] of a quiver Q= •→•→• is injective if and only if each M₁,M₂,M₃ is injective left R-module and ker(f₁) is a summand of M₁ and ker(f₂) is a summand of M₂.
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