Exercise 10.3.2. Let R be a commutative ring with identity. For all positive integers n and m, R ∼= R if and only if n = m. Proof. Let φ : R → R be an isomorphism of R-modules and let I E R be a maximal ideal. Then the map φ̄ : R → R/IR given by φ̄(α) = φ(α) is a morphism of R-modules. Moreover ker φ̄ = {α ∈ R | φ̄(α) = 0} = {α ∈ R | φ(α) ∈ IR} = φ−1(IRm) = IR. Therefore by the first isomorphism th...

This paper is devoted to introduce and investigate some weak forms of fuzzy open mappings, namely fuzzy faintly semi open (fuzzy faintly semi closed), fuzzy faintly preopen (fuzzy faintly preclosed), fuzzy faintly α-open (fuzzy faintly α-closed), fuzzy faintly semi preopen (fuzzy faintly semi preclosed) and fuzzy faintly spopen (fuzzy faintly spclosed) mappings and their fundamental properties ...

Bosonized q-vertex operators related to the 4-dimensional evaluation modules of the quantum affine superalgebra Uq[ ̂ sl(2|1)] are constructed for arbitrary level k = α, where α 6= 0,−1 is a complex parameter appearing in the 4-dimensional evaluation representations. They are intertwiners among the level-α highest weight Fock-Wakimoto modules. Screen currents which commute with the action of Uq[...

We introduce an axiomatic extension of Höhle’s Monoidal Logic called Semi–divisible Monoidal Logic, and prove that it is complete by showing that semi–divisibility is preserved in MacNeille completion. Moreover, we introduce Strong semi– divisible Monoidal Logic and conjecture that a predicate formula α is derivable in Strong Semi–divisible Monadic logic if, and only if its double negation ¬¬α ...

The purpose of this paper is to outline a new approach to the classii-cation of nitely generated indecomposable modules over certain kinds of pullback rings. If R is the pullback of two hereditary noetherian serial rings over a common semi{simple artinian ring, then this classiication can be divided into the classiica-tion of indecomposable artinian modules and those modules over the coordinate...

In this paper, as the first step towards classification of simple weight modules with finite dimensional weight spaces over Witt algebras Wn, we explicitly describe supports of such modules. We also obtain some descriptions on the support of an arbitrary simple weight module over a Z-graded Lie algebra g having a root space decomposition ⊕α∈Zngα with respect to the abelian subalgebra g0, with t...

In this paper, as the first step towards classification of simple weight modules with finite dimensional weight spaces over Witt algebras Wn, we explicitly describe supports of such modules. We also obtain some descriptions on the support of an arbitrary simple weight module over a Z-graded Lie algebra g having a root space decomposition ⊕α∈Zngα with respect to the abelian subalgebra g0, with t...

Quantized enveloping algebras have been widely studied, almost exclusively by theoretical means (see, for example, De Concini and Procesi, 1993; Jantzen, 1996; Lusztig, 1993). In this paper we consider the problem of computing with a quantized enveloping algebra. For this we need a basis of it, along with a method for computing the product of two basis elements. To this end we will use so-calle...

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(1.1) 0 −→ N f −−→M g −−→ P −→ 0 which is exact at N , M , and P . That means f is injective, g is surjective, and im f = ker g. Example 1.1. For an R-module M and submodule N , there is a short exact sequence 0 // N // M // M/N // 0, where the map N →M is the inclusion and the map M →M/N is reduction modulo N . Example 1.2. For R-modules N and P , the direct sum N ⊕ P fits into the short exact...