Auslander Algebra Trivia Quiz

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Auslander Algebra Trivia Quiz - Quiz

The Auslander Algebra is a work of Maurice Auslander a famous American Mathematician. His work was an Artin Algebra representation. An Artin Algebra is an Auslander Algebra if gl dim Γ ≤ 2 and if 0→Γ→I→J→K→0 is a minimal injective resolution of Γ then I and J are projective Γ-modules; where Γ is the Artin Algebra. Try out this quiz.


Questions and Answers
  • 1. 

    Which of these is a summed up module of Algebra A?

    • A.

      Integrated modules

    • B.

      Indecomposable module

    • C.

      Projective module

    • D.

      Isomorphic module

    Correct Answer
    B. Indecomposable module
    Explanation
    An indecomposable module is a module that cannot be expressed as the direct sum of two non-zero submodules. In other words, it cannot be "broken down" into smaller modules. This makes it a summed up module of Algebra A because it represents a fundamental building block that cannot be further simplified or decomposed.

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  • 2. 

    When did Auslander introduce the Algebra?

    • A.

      1874

    • B.

      1870

    • C.

      1974

    • D.

      1989

    Correct Answer
    C. 1974
  • 3. 

    In these fomular, k[T ]=hT n i, what does k stand for?

    • A.

      Term

    • B.

      Natural value

    • C.

      Module

    • D.

      Field

    Correct Answer
    D. Field
    Explanation
    The correct answer is "Field". In the given formula, k represents the field. A field is a mathematical structure that consists of a set of elements along with operations such as addition and multiplication. In this context, k is the field used to represent the relationship between temperature (T) and the rate constant (k) in a chemical reaction.

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  • 4. 

    What does T represent in the Auslander formular?

    • A.

      Constant

    • B.

      Value for Artin constant

    • C.

      Variable

    • D.

      Function

    Correct Answer
    C. Variable
    Explanation
    The correct answer is "Variable." In the Auslander formula, T represents a variable. This means that its value can change and is not fixed.

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  • 5. 

    What does the n in the Auslander formular stand for?

    • A.

      Integers

    • B.

      Natural Number

    • C.

      Rational number

    • D.

      Constant

    Correct Answer
    B. Natural Number
    Explanation
    The "n" in the Auslander formula stands for natural number. A natural number is a positive integer greater than zero that is used to count or label objects. In the context of the Auslander formula, it is likely used as a parameter or variable representing a specific natural number value.

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  • 6. 

    Which of these symbol represents Artin Algebra in relation to Auslander Algebra?

    • A.

    • B.

      μ

    • C.

      Λ

    • D.

      Γ

    Correct Answer
    D. Γ
    Explanation
    The symbol Γ represents Artin Algebra in relation to Auslander Algebra. This is because Γ is commonly used to denote the Auslander-Reiten quiver, which is a directed graph that represents the indecomposable modules of an Artin algebra. The Auslander-Reiten quiver provides important information about the structure and representation theory of the Artin algebra, making Γ the correct symbol to represent it.

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  • 7. 

    What kind of ring is the sum of indecomposable module of A in the Auslander Algebra?

    • A.

      Module

    • B.

      Endomorphism

    • C.

      Homomorphism

    • D.

      Artin

    Correct Answer
    B. Endomorphism
    Explanation
    The sum of indecomposable modules of A in the Auslander Algebra is an endomorphism. This means that it is a module that is both a domain and a codomain of a homomorphism, and the homomorphism is from the module to itself. In other words, it represents the set of all possible mappings from the module to itself.

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  • 8. 

    Which of these satisfies the condition for an Artin Algebra to be an Auslander Algebra?

    • A.

      Γ ≤ 2

    • B.

      Γ ≥ 2

    • C.

      Γ < 1

    • D.

      Γ > 2

    Correct Answer
    A. Γ ≤ 2
    Explanation
    An Artin Algebra is an Auslander Algebra if its global dimension (Γ) is less than or equal to 2. This means that the algebra has a finite projective resolution of length 2 or less.

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  • 9. 

    Which of these people is connected to the Auslander Algebra?

    • A.

      Reinten

    • B.

      Becleau

    • C.

      Herin

    • D.

      Phills

    Correct Answer
    A. Reinten
    Explanation
    Reinten is connected to the Auslander Algebra.

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  • 10. 

    Which of these explains Auslander Algebra of any representation of finite Algebra?

    • A.

      Budding of Fl

    • B.

      Blossom of flowers

    • C.

      Quasi Hereditary

    • D.

      Passing of Gene

    Correct Answer
    C. Quasi Hereditary
    Explanation
    Quasi Hereditary is an explanation for the Auslander Algebra of any representation of a finite Algebra. Quasi Hereditary refers to a property of certain categories that allows for the decomposition of modules into a direct sum of indecomposable modules. This property is closely related to the Auslander-Reiten theory, which studies the structure of modules and their relations in a given algebra. Therefore, the concept of Quasi Hereditary provides a valid explanation for the Auslander Algebra of any representation of a finite Algebra.

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  • Current Version
  • Mar 18, 2023
    Quiz Edited by
    ProProfs Editorial Team
  • Apr 09, 2019
    Quiz Created by
    Gregorynaomi

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