Let R and S be rings, and let φ: R →S be a ring homomorphism. If M is a left S-module, prove tha

step 1 In the question, they are asking that let P be a prime and M be a positive integer. By mathemati

Let R and S be rings, and let φ: R →S be a ring...

Let $R$ and $S$ be rings, and let $\varphi: R \rightarrow S$ be a ring homomorphism. If $M$ is a left $S$-module, prove that $M$ is also a left $R$-module if we define
$$
r m=\varphi(r) m,
$$
for all $r \in R$ and $m \in M$.

Key Concepts

Ring

A ring is an algebraic structure consisting of a set equipped with two binary operations—addition and multiplication—that satisfy properties such as associativity, distributivity of multiplication over addition, the existence of an additive identity, and the existence of additive inverses. In many contexts, rings also have a multiplicative identity, forming a unital ring.

Module

A module is a generalization of the concept of a vector space where the scalars come from a ring rather than a field. It consists of an abelian group together with an operation of scalar multiplication that is compatible with the ring operations, thereby satisfying axioms like distributivity and associativity of scalar multiplication with respect to the ring multiplication.

Ring Homomorphism

A ring homomorphism is a function between two rings that preserves the ring operations, meaning it respects both the addition and multiplication structures of the rings. That is, it satisfies the conditions that the image of the sum (or product) of any two elements equals the sum (or product) of their images, and if the rings are unital, it maps the multiplicative identity of the source ring to that of the target ring.

Restriction of Scalars

Restriction of scalars is a process by which a module over one ring is regarded as a module over another ring via a ring homomorphism. Specifically, if there is a ring homomorphism from one ring to another, the scalar multiplication in the module defined over the target ring can be reinterpreted to define an action of the source ring, thereby endowing the module with a compatible structure over the source ring.

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