In this paper, we study the left orthogonal class of max-flat modules which are homological objects related to s-pure exact sequences and module homomorphisms. Namely, a right $A$ is called MF-projective if ${Ext}^{1}_{R}(A,B)=0$ for any $R$-module $B$, strongly ${Ext}^{i}_{R}(A,B)=0$ all $R$-modules $B$ $i\geq 1$. Firstly, give some properties $MF$-projective SMF-projective modules. Then intro...

We show that any stack X of finite type over a Noetherian scheme has presentation X→X by such X(F)→X(F) is onto, for every or real closed field F. Under some additional conditions on X, we the same all perfect fields. prove similar results (some) Henselian rings. give two applications main result. One to counting isomorphism classes stacks rings Z/pn; other about relation between algebraic and ...

we show  that a projective maximal submodule of afinitely generated, regular, extending module is a directsummand. hence, every finitely generated, regular, extendingmodule with projective maximal submodules is semisimple. as aconsequence, we observe that every regular, hereditary, extendingmodule is semisimple. this generalizes and simplifies a result of  dung and   smith. as another consequen...

Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theorem. We give a new short proof of the main case of this lemma. In this note, we also give a short proof of Hayward’s decomposition theorem for weakly chordal graphs, relying on a Roussel–Rubio-type lemma. We recall how Roussel–Rubio-type lemmas yield very short proofs of the existence of even pairs ...

We will extend Reed's Semi-Strong Perfect Graph Theorem by proving that unbreakable C 5-free graphs diierent from a C 6 and its complement have unique P 4-structure.

Persistent current and low-field magnetic susceptibility in single-channel normal metal rings threaded by a magnetic flux φ are investigated within the tight-binding framework considering long-range hopping of electrons in the shortest path. The higher order hopping integrals try to reduce the effect of disorder by delocalizing the energy eigenstates, and accordingly, current amplitude in disor...

Persistent current and low-field magnetic susceptibility in single-channel normal metal rings threaded by a magnetic flux φ are investigated within the tight-binding framework considering long-range hopping of electrons in the shortest path. The higher order hopping integrals try to reduce the effect of disorder by delocalizing the energy eigenstates, and accordingly, current amplitude in disor...

We obtain a characterization of left perfect rings via superstability the class flat modules with pure embeddings. $\mathbf{Theorem.}$ For ring $R$ following are equivalent. - is perfect. The $R$-modules embeddings superstable. There exists $\lambda \geq (|R| + \aleph_0)^+$ such that has uniqueness limit models cardinality $\lambda$. Every model in $\Sigma$-cotorsion. A key step our argum...

We give a refinement of Saito's arithmetic duality for two-dimensional local rings by giving algebraic group structures cohomology groups.