What Do You Know About Shapiro's Lemma?

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What Do You Know About Shapiros Lemma? - Quiz

Also known as the Eckmann–Shapiro Lemma, Shapiro's lemma is useful in numerous areas of the abstract algebra field in mathematics.
In the 21st century, Beno Eckmann and Arnold Shapiro individually contributed to the realization of this concept, which relates extensions of modules over one ring to extensions over another.
This especially works best in the group ring of a group and of a subgroup.


Questions and Answers
  • 1. 

    Shapiro's lemma is also known as which of these?

    • A.

      Eckmann Lemma

    • B.

      Eckmann—Shapiro Lemma

    • C.

      Erick—Shapiro Lemma

    • D.

      Erick Lemma

    Correct Answer
    B. Eckmann—Shapiro Lemma
    Explanation
    Shapiro's lemma is commonly known as Eckmann-Shapiro Lemma.

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  • 2. 

    Who proved the theorem? 

    • A.

      Arnold Shapiro

    • B.

      Alfred Shapiro

    • C.

      Josh Shapiro

    • D.

      Matt Shapiro

    Correct Answer
    A. Arnold Shapiro
  • 3. 

    When was the theorem proven?

    • A.

      1961

    • B.

      1962

    • C.

      1963

    • D.

      1964

    Correct Answer
    A. 1961
    Explanation
    The theorem was proven in 1961.

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  • 4. 

     Who first discovered the theorem? 

    • A.

      Beno Erick

    • B.

      Alfred Eckmann

    • C.

      Beno Eckmann

    • D.

      Alfred Erick

    Correct Answer
    C. Beno Eckmann
    Explanation
    Beno Eckmann is the correct answer because he was the first person to discover the theorem.

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  • 5. 

    When was the theorem discovered? 

    • A.

      1951

    • B.

      1952

    • C.

      1953

    • D.

      1954

    Correct Answer
    C. 1953
    Explanation
    The theorem was discovered in 1953.

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  • 6. 

    What does R → S  denote?

    • A.

      Ring homomorphism

    • B.

      Ring endomorphism

    • C.

      Ring

    • D.

      Ring and Satellite

    Correct Answer
    A. Ring homomorphism
    Explanation
    R -> S denotes a ring homomorphism. A ring homomorphism is a function between two rings R and S that preserves the ring structure, meaning it preserves addition, multiplication, and the identity element. This means that if we have two elements a and b in R, their sum and product in R will be preserved under the homomorphism.

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  • 7. 

    What does M denote?

    • A.

      Right S-module

    • B.

      Left S-module

    • C.

      Left M-module

    • D.

      Right M-module

    Correct Answer
    B. Left S-module
    Explanation
    The correct answer is "Left S-module" because in module theory, the term "left" refers to the side on which the module acts. Since M is denoted as a "Left S-module," it implies that M is a module that acts on the left side with respect to the ring S.

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  • 8. 

    What does N denote?

    • A.

      Left R-module

    • B.

      Right R-module

    • C.

      Left N-module

    • D.

      Right N-module

    Correct Answer
    A. Left R-module
    Explanation
    N denotes a module over the ring R that is acted upon by elements of R on the left. In other words, N is a module where the scalar multiplication is defined by multiplying elements of R on the left with elements of N. This is in contrast to a right R-module, where the scalar multiplication is defined by multiplying elements of R on the right with elements of N.

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  • 9. 

    What does S denote?

    • A.

      Left and right S-module

    • B.

      Left and right R-module

    • C.

      Left and right M-module

    • D.

      Left and right N-module

    Correct Answer
    B. Left and right R-module
    Explanation
    The correct answer is "Left and right R-module" because the letter "S" typically denotes a module over a ring "R." In this context, "R" represents a ring and "S" represents a module that can be both a left module and a right module over the ring "R."

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  • 10. 

    What does R(G) denote?

    • A.

      Group of modules

    • B.

      Radius

    • C.

      Ring

    • D.

      Group ring

    Correct Answer
    D. Group ring
    Explanation
    R(G) denotes the group ring. In mathematics, the group ring is a construction that associates to every group G a ring RG, whose elements are formal finite linear combinations of elements of G with coefficients in a ring R. The group ring is an important concept in algebra and has applications in various areas of mathematics, such as representation theory and algebraic topology.

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  • Current Version
  • Mar 18, 2023
    Quiz Edited by
    ProProfs Editorial Team
  • Jun 18, 2018
    Quiz Created by
    AdeKoju
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