# What Do You Know About Shapiro's Lemma?

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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.

• 1.

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

• A.

Eckmann Lemma

• B.

Eckmann—Shapiro Lemma

• C.

Erickâ€”Shapiro Lemma

• D.

Erick Lemma

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

A. Arnold Shapiro
• 3.

### When was the theorem proven?

• A.

1961

• B.

1962

• C.

1963

• D.

1964

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

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

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

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

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

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

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.

• C.

Ring

• D.

Group ring

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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• Mar 18, 2023
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• Jun 18, 2018
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