let r be a right gf-closed ring with finite left and right gorenstein global dimension. we prove that if i is an ideal of r such that r/i is a semi-simple ring, then the gorensntein flat dimensnion of r/i as a right r-module and the gorensntein injective dimensnnion of r/i as a left r-module are identical. in particular, we show that for a simple module s over a commutative gorensntein ring r, ...

We characterize the functions f : ( 0 , 1 ] ⟶ [ for which there exists a measurable set C ⊆ of positive measure satisfying | ∩ I < ) any nontrivial interval . As an application, we prove that on injective curve it is possible to construct Lipschitz function cannot be approximated by attaining their constant. Finally, extend this result more general curves.

Given two graphs G = (VG, EG) and H = (VH , EH), a homomorphism from G to H is a function f : VG → VH such that for every uv ∈ EG, f(u)f(v) ∈ EH . A homomorphism from G to H is referred to as an H-colouring of G and the vertices of H are regarded as colours. The graph H is called the target of the homomorphism. These definitions extend to directed graphs by requiring that the mapping must prese...

Let us first examine this sequence in the 1-dimensional case. When the underlying manifold is C, the claim is that 0→ Hol→ C∞ ∂ −→ Ω → 0 is exact. Clearly, Hol → C∞ is injective, and we observed last class that Hol is precisely the kernel of the map ∂ : C∞ → Ω0,1, so the only difficulty in verifying exactness is in showing that ∂ : C∞ → Ω0,1 is surjective. That is, given a smooth function g : C...

We define the concepts of an injective colouring and the injective chromatic number of a graph and give some upper and lower bounds in general, plus some exact values. We explore in particular the injective chromatic number of the hypercube and put it in the context of previous work on similar concepts, especially the theory of errorcorrecting codes. Finally, we give necessary and sufficient co...

We prove that, except in some low-complexity cases, every locally injective simplicial map between pants graphs is induced by a π1-injective embedding between the corresponding surfaces.

Let [Formula: see text] be a ring, class of text]-modules and an integer. We introduce the concepts Gorenstein text]-[Formula: text]-injective text]-flat modules via special finitely presented modules. Besides, we obtain some equivalent properties these on text]-coherent rings. Then investigate relations among text]-injective, text]-flat, injective flat text]-rings (i.e., self rings). Several k...

If R̂ is the pure-injective hull of a valuation ring R, it is proved that R̂ ⊗R M is the pure-injective hull of M , for every finitely generated Rmodule M . Moreover R̂ ⊗R M ∼= ⊕1≤k≤nR̂/AkR̂, where (Ak)1≤k≤n is the annihilator sequence of M . The pure-injective hulls of uniserial or polyserial modules are also investigated. Any two pure-composition series of a countably generated polyserial module a...