What Do You Know About Shapiro's Lemma?

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.

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.

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.

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."

Shapiro's lemma is commonly known as Eckmann-Shapiro Lemma.

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Beno Eckmann is the correct answer because he was the first person to discover the theorem.

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.

The theorem was proven in 1961.

The theorem was discovered in 1953.