INJECTIVE AND PROJECTIVE PROPERTIES OF REPRESENTATIONS OF QUIVERS WITH n EDGES
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We de¯ne injective and projective representations of quivers with two vertices with n arrows. In the representation of quivers we denote n edges between two vertices as ) and n maps as f1 » fn, and E ©E ©¢ ¢ ¢©E (n times) as ©nE. We show that if E is an injective left R-module, then ©nE p1spn+3E is an injective representation of Q = ²)² where pi(a1; a2; ¢ ¢ ¢ ; an) = ai, i 2 f1; 2; ¢ ¢ ¢ ; ng. Dually we show that if M1 f1sfn+3M2 is an injective representation of a quiver Q = ²)² then M1 and M2 are injective left R-modules. We also show that if P is a projective left R-module, then P i1sin +3©nP is a projective representation of Q = ²)² where ik is the kth in- jection. And if M1 f1sfn+3M2 is an projective representation of a quiver Q = ²)² then M1 and M2 are projective left R-modules.
Abstract
We de¯ne injective and projective representations of quivers with two vertices with n arrows. In the representation of quivers we denote n edges between two vertices as ) and n maps as f1 » fn, and E ©E ©¢ ¢ ¢©E (n times) as ©nE. We show that if E is an injective left R-module, then ©nE p1spn+3E is an injective representation of Q = ²)² where pi(a1; a2; ¢ ¢ ¢ ; an) = ai, i 2 f1; 2; ¢ ¢ ¢ ; ng. Dually we show that if M1 f1sfn+3M2 is an injective representation of a quiver Q = ²)² then M1 and M2 are injective left R-modules. We also show that if P is a projective left R-module, then P i1sin +3©nP is a projective representation of Q = ²)² where ik is the kth in- jection. And if M1 f1sfn+3M2 is an projective representation of a quiver Q = ²)² then M1 and M2 are projective left R-modules.
참고문헌 (7)
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