What Do You Know About Auslander—reiten Theory?

Approved & Edited by ProProfs Editorial Team
The editorial team at ProProfs Quizzes consists of a select group of subject experts, trivia writers, and quiz masters who have authored over 10,000 quizzes taken by more than 100 million users. This team includes our in-house seasoned quiz moderators and subject matter experts. Our editorial experts, spread across the world, are rigorously trained using our comprehensive guidelines to ensure that you receive the highest quality quizzes.
Learn about Our Editorial Process
| By AdeKoju
A
AdeKoju
Community Contributor
Quizzes Created: 129 | Total Attempts: 38,844
Questions: 10 | Attempts: 125

SettingsSettingsSettings
What Do You Know About Auslanderreiten Theory? - Quiz

When we reunite broken parts (i. E. In algebra), the Auslander–Reiten theory becomes a very useful theory, which —
thanks to specific techniques — studies the representation theory of Artinian rings.
These techniques include Auslander–Reiten quivers, Auslander–Reiten sequences, and almost split sequences.
And as the name implies, mathematicians Reiten and Auslander introduced it in the 20th century.


Questions and Answers
  • 1. 

     When was Auslander–Reiten theory introduced?

    • A.

      1975

    • B.

      1976

    • C.

      1977

    • D.

      1978

    Correct Answer
    A. 1975
    Explanation
    Auslander-Reiten theory was introduced in 1975.

    Rate this question:

  • 2. 

    What does the theory determine?

    • A.

      Representation theory of Artinian rings

    • B.

      Auslander theory of Artinian rings

    • C.

      Independent theory of Artinian rings

    • D.

      Artinian rings

    Correct Answer
    A. Representation theory of Artinian rings
    Explanation
    The theory being referred to in the given question is the representation theory of Artinian rings. This theory focuses on the study of how elements of Artinian rings can be represented as linear transformations on vector spaces. It explores the relationship between the algebraic structure of Artinian rings and their corresponding linear transformations, providing insights into the behavior and properties of these rings.

    Rate this question:

  • 3. 

    What is another name for Auslander—Reiten sequence?

    • A.

      Almost split sequence

    • B.

      Split sequence

    • C.

      Sectional sequence

    • D.

      Total sequence

    Correct Answer
    A. Almost split sequence
    Explanation
    The correct answer is "Almost split sequence." The Auslander-Reiten sequence is also known as the almost split sequence. This sequence is a fundamental tool in the study of representation theory and homological algebra. It provides important information about the structure and properties of modules over certain algebraic structures, such as rings or algebras. The almost split sequence helps to understand the relationships between different modules and their homological properties, making it a crucial concept in these areas of mathematics.

    Rate this question:

  • 4. 

    Which duo introduced the theory? 

    • A.

      Maurice Auslander and Idun Reiten

    • B.

      Idun Auslander and Maurice Reiten

    • C.

      Leo Auslander and Idun Reiten

    • D.

      Maurice Auslander and Charles Reiten

    Correct Answer
    A. Maurice Auslander and Idun Reiten
    Explanation
    Maurice Auslander and Idun Reiten introduced the theory.

    Rate this question:

  • 5. 

    What is the English expression of τ = D Tr?

    • A.

      Transition

    • B.

      Transmission

    • C.

      Translation

    • D.

      Transpose

    Correct Answer
    A. Transition
    Explanation
    The English expression of τ = D Tr is "Transition". In mathematics, τ represents a transition matrix, D represents a diagonal matrix, and Tr represents the trace of a matrix. Therefore, the equation can be read as "Transition matrix equals the product of a diagonal matrix and the trace of a matrix".

    Rate this question:

  • 6. 

     In τ = D Tr, what is D? 

    • A.

      Diameter

    • B.

      Dual

    • C.

      Density

    • D.

      Distance

    Correct Answer
    B. Dual
    Explanation
    In the equation τ = D Tr, the variable D represents the concept of "dual". The equation suggests that the value of τ is equal to D multiplied by Tr. Therefore, D is the coefficient or factor associated with the concept of "dual" in this equation.

    Rate this question:

  • 7. 

    What is Tr equal to?

    • A.

      Transition

    • B.

      Transpose

    • C.

      Transmission

    • D.

      Translation

    Correct Answer
    B. Transpose
    Explanation
    The correct answer is "Transpose." Transpose refers to the operation of interchanging rows and columns in a matrix. It is commonly used in linear algebra and mathematics to manipulate matrices and solve equations.

    Rate this question:

  • 8. 

    Which of these is a property of Auslander—Reiten sequence? 

    • A.

      The sequence is split

    • B.

      The sequence is not split

    • C.

      The sequence is almost split

    • D.

      The sequence remains in the same form

    Correct Answer
    B. The sequence is not split
    Explanation
    The property of Auslander-Reiten sequence being "not split" means that the sequence does not have a direct summand that is isomorphic to the original module. In other words, the sequence cannot be decomposed into simpler modules that are isomorphic to the original module. This property is important in the study of representation theory and module theory, as it provides information about the structure and behavior of modules.

    Rate this question:

  • 9. 

     Which ring is used to determine the theory?

    • A.

      Artic ring

    • B.

      Artinian ring

    • C.

      Attic ring

    • D.

      Indecompossable ring

    Correct Answer
    B. Artinian ring
    Explanation
    The correct answer is Artinian ring. An Artinian ring is a ring in which every descending chain of ideals stabilizes, meaning that there are no infinitely descending chains of ideals. This property is important in the study of ring theory as it allows for the development of various results and theorems. The other options (Artic ring, Attic ring, and Indecompossable ring) are not commonly used terms in ring theory and do not have the same significance as the Artinian ring.

    Rate this question:

  • 10. 

    In the theory, what does k represent?

    • A.

      Field

    • B.

      Kelvin value

    • C.

      Valency

    • D.

      Velocity

    Correct Answer
    A. Field
    Explanation
    In the theory, k represents the field.

    Rate this question:

Quiz Review Timeline +

Our quizzes are rigorously reviewed, monitored and continuously updated by our expert board to maintain accuracy, relevance, and timeliness.

  • Current Version
  • Mar 20, 2023
    Quiz Edited by
    ProProfs Editorial Team
  • Jun 17, 2018
    Quiz Created by
    AdeKoju

Related Topics

Back to Top Back to top
Advertisement
×

Wait!
Here's an interesting quiz for you.

We have other quizzes matching your interest.