Let [Formula: see text] be a ring and [Formula: see text] be a finite quiver. We give an explicit formula for the injective envelopes and projective precovers in the category [Formula: see text] of bound representations of [Formula: see text] by left [Formula: see text]-modules, where [Formula: see text] is a combination of monomial and commutativity relations. Some applications will be provided. In particular, it is shown that if [Formula: see text] is acyclic and [Formula: see text] is an Iwanaga-Gorenstein ring, then so are these bound quiver algebras.
In this research two grades of polysulfide resin with low and high molecular weight (respectively G4 and G112) as reactive modifier was used to toughen epoxy resin. The effect of modifier molecular weight on impact resistance, thermal expansion coefficient, storage and loss modulus, decomposition temperature and adhesion properties of toughened epoxy was investigated. The impact strength and the thermal expansion coefficient (CTE) of epoxy resin was increased with increasing polysulfide but the G112 modified epoxy samples showed higher CTE values and impact resistance than those of modified with G4. Comparing of the same weight percent inclusion of G4 and G112 effect on decomposition temperature show that G4 modified epoxy resin has lower decomposition temperature than the G112 modified epoxy resin. Also addition of G112 up to 10 weight percent leads to higher bond strength with aluminum sheets. According to the DMTA graphs, glass transition temperature (Tg) of the modified epoxy was decreased with increasing polysulfide weight percent in composition. At the same time G4 modified epoxies have lower Tg and storage modulus than that of modified with G112.
Let R be a ring and Q be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category Rep(Q, R) of representations of Q by left R-modules. We also extend our formula to all terms of the minimal injective resolution of RQ. Using such descriptions, we study the Auslander-Gorenstein property of path algebras. In particular, we prove that the path algebra RQ is k-Gorenstein if and only if Q = − → An and R is a k-Gorenstein ring, where n is the number of vertices of Q.
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