Introduction 1 0.1. Some theorems in mathematics with snappy model-theoretic proofs 1 1. Languages, structures, sentences and theories 1 1.1. Languages 1 1.2. Statements and Formulas 4 1.3. Satisfaction 5 1.4. Elementary equivalence 6 1.5. Theories 7 2. Big Theorems: Completeness, Compactness and Löwenheim-Skolem 9 2.1. The Completeness Theorem 9 2.2. Proof-theoretic consequences of the complet...

For a commutative noetherian ring A, we compare the support of a complex of A-modules with the support of its cohomology. This leads to a classification of all full subcategories of A-modules which are thick (that is, closed under taking kernels, cokernels, and extensions) and closed under taking direct sums.

We introduce image-cokernel-extension-closed (ICE-closed) subcategories of module categories. This class unifies both torsion classes and wide subcategories. show that ICE-closed over the path algebra Dynkin type are in bijection with basic rigid modules precisely some also study natural maps from to terms modules. Finally, we prove number does not depend on orientation quiver, give an explicit...

let $r$ be a commutative noetherian ring with non-zero identity, $fa$ an ideal of $r$, and $x$ an $r$--module. here, for fixed integers $s, t$ and a finite $fa$--torsion $r$--module $n$, we first study the membership of $ext^{s+t}_{r}(n, x)$ and $ext^{s}_{r}(n, h^{t}_{fa}(x))$ in the serre subcategories of the category of $r$--modules. then, we present some conditions which ensure the exi...

The objective of this study is to investigate the fuzzy subhypernear-modules over hypernear-ring by using a triangular norm, which T-fuzzy are generalization submodules hyper near-modules, and some related properties discussed. Idempotent subhypernear-module hypernear-module suggested. relation between level subset presented. Also, image inverse under mapping introduced. direct product i...

Let $R$ be a commutative Noetherian ring and let $fa$, $fb$ be two ideals of $R$ such that $R/({fa+fb})$ is Artinian. Let $M$, $N$ be two finitely generated $R$-modules. We prove that $H_{fb}^j(H_{fa}^t(M,N))$ is Artinian for $j=0,1$, where $t=inf{iin{mathbb{N}_0}: H_{fa}^i(M,N)$ is not finitelygenerated $}$. Also, we prove that if $DimSupp(H_{fa}^i(M,N))leq 2$, then $H_{fb}^1(H_{fa}^i(M,N))$ i...