Projective properties of representations of a quiver Q= ● → ● as R[x]-modules
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In this paper we extend the projective properties of representations of a quiver Q= ● → ● as left R-modules to the projective properties of representations of quiver Q= ● → ● as left R[x]-modules. We show that if P is a projective left R-module then 0 → P[x] is a projective representation of a quiver Q = ● → ●as R[x]-modules. And we show 0 → L is a projective representation of Q = ● → ● as R-module if and only if 0 → L[x] is a projective representation of a quiver Q = ● → ● as R[x]-modules. Then we show if P is a projective left R-module then P[x] →id P[x] is a projective representation of a quiver Q ● → ● as R[x]-modules. We also show that if L →id L is a projective representation of Q = ● → ● as R-module, then L[x]→id L[x] is a projective representation of a quiver Q =● → ● as R[x]-modules.
Abstract
In this paper we extend the projective properties of representations of a quiver Q= ● → ● as left R-modules to the projective properties of representations of quiver Q= ● → ● as left R[x]-modules. We show that if P is a projective left R-module then 0 → P[x] is a projective representation of a quiver Q = ● → ●as R[x]-modules. And we show 0 → L is a projective representation of Q = ● → ● as R-module if and only if 0 → L[x] is a projective representation of a quiver Q = ● → ● as R[x]-modules. Then we show if P is a projective left R-module then P[x] →id P[x] is a projective representation of a quiver Q ● → ● as R[x]-modules. We also show that if L →id L is a projective representation of Q = ● → ● as R-module, then L[x]→id L[x] is a projective representation of a quiver Q =● → ● as R[x]-modules.
참고문헌 (8)
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