Projective representations of a quiver with three vertices and two edges as $R[x]$-modules
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In this paper we show that the projective properties of representations of a quiver Q = • → • → • as left R[x]-modules. We show that if P is a projective left R-module then 0 → 0 → P[x]is a projective representation of a quiver Q as R[x]-modules, but P[x] → 0 →0 is not a projective representation of a quiver Q as R[x]-modules, if P ̸= 0. And we show a representation 0 / P[x](→) id P[x] of a quiver Q is a projective representation,if P is a projective left R-module, but P[x]( →)id /P[x] → 0 is not a projective representation of a quiver Q as R[x]-modules, if P ̸=0. Then we show a representation P[x] (→)id P[x](→) id P[x] of a quiver Q is a projective representation, if P is a projective left R-module.
Abstract
In this paper we show that the projective properties of representations of a quiver Q = • → • → • as left R[x]-modules. We show that if P is a projective left R-module then 0 → 0 → P[x]is a projective representation of a quiver Q as R[x]-modules, but P[x] → 0 →0 is not a projective representation of a quiver Q as R[x]-modules, if P ̸= 0. And we show a representation 0 / P[x](→) id P[x] of a quiver Q is a projective representation,if P is a projective left R-module, but P[x]( →)id /P[x] → 0 is not a projective representation of a quiver Q as R[x]-modules, if P ̸=0. Then we show a representation P[x] (→)id P[x](→) id P[x] of a quiver Q is a projective representation, if P is a projective left R-module.
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