Question

Suppose \(M_{1}, \ldots, M_{k}\) are \(R\) -modules. Show that for each \(i=\) \(1, \ldots, k,\) the projection map \(\pi_{i}: M_{1} \times \cdots \times M_{k} \rightarrow M_{i}\) that sends \(\left(\alpha_{1}, \ldots, \alpha_{k}\right)\) to \(\alpha_{i}\) is a surjective \(R\) -linear map.

Short answer

Question: Prove that for each \(i\), the projection map \(\pi_i: M_1 \times \dots \times M_k \rightarrow M_i\) is a surjective \(R\)-linear map, given \(k\) \(R\)-modules \(M_1, \dots, M_k\). Solution: 1. We first reviewed the definitions of an \(R\)-linear map and a surjective function. 2. Then, we proved that the projection map \(\pi_i\) is an \(R\)-linear map by showing it satisfies the two properties of an \(R\)-linear map. 3. Finally, we showed that \(\pi_i\) is surjective by proving that for every element \(\alpha_i\) in \(M_i\), there exists a tuple in \(M_1 \times \dots \times M_k\) that maps to \(\alpha_i\) under \(\pi_i\). Therefore, we have proved that for each \(i\), the projection map \(\pi_i: M_1 \times \dots \times M_k \rightarrow M_i\) is a surjective \(R\)-linear map.

Step-by-step solution

Review Definitions of \(R\)-linear Map and Surjective Function

First, let's review the definitions of an \(R\)-linear map and a surjective function: - An \(R\)-linear map \(f: M \rightarrow N\) is a function between two \(R\)-modules that satisfies the following properties: 1. \(f(\alpha + \beta) = f(\alpha) + f(\beta)\) for all \(\alpha, \beta \in M\). 2. \(f(r\alpha) = rf(\alpha)\) for all \(r \in R\) and \(\alpha \in M\). - A function \(f: A \rightarrow B\) is called surjective (also known as onto) if for every element \(b \in B\), there exists at least one element \(a \in A\) such that \(f(a) = b\). Now we are ready to prove that the projection map \(\pi_i\) is an \(R\)-linear map and surjective.

Prove \(\pi_i\) is an \(R\)-linear Map

We need to show that \(\pi_i\) is an \(R\)-linear map, which means proving it satisfies the two properties listed in Step 1. Let \(\left(\alpha_1, \dots, \alpha_k\right), \left(\beta_1, \dots, \beta_k\right) \in M_1 \times \dots \times M_k\), and \(r \in R\). We have these cases: 1. \(i\)-th component addition: \(\pi_i(\left(\alpha_1, \dots, \alpha_k\right) + \left(\beta_1, \dots, \beta_k\right)) = \pi_i(\alpha_1 + \beta_1, \dots, \alpha_k + \beta_k) = \alpha_i + \beta_i = \pi_i(\left(\alpha_1, \dots, \alpha_k\right)) + \pi_i(\left(\beta_1, \dots, \beta_k\right))\). 2. Scalar multiplication: \(\pi_i\left(r(\alpha_1, \dots, \alpha_k)\right) = \pi_i(r\alpha_1, \dots, r\alpha_k) = r\alpha_i = r \pi_i(\left(\alpha_1, \dots, \alpha_k\right))\). Since \(\pi_i\) satisfies both of the properties for an \(R\)-linear map, \(\pi_i\) is an \(R\)-linear map.

Prove \(\pi_i\) is Surjective

Now we need to show that \(\pi_i\) is surjective. In other words, we need to prove that for every element \(\alpha_i \in M_i\), there is at least one element in \(M_1 \times \dots \times M_k\) that maps to \(\alpha_i\) under \(\pi_i\). Let \(\alpha_i \in M_i\). Consider the tuple \(\left(\alpha_{1,0}, \dots, \alpha_{i-1,0}, \alpha_i, \alpha_{i+1,0}, \dots, \alpha_{k,0}\right) \in M_1 \times \dots \times M_k\), where \(\alpha_{j,0}\) is the zero element of \(M_j\) for \(j \neq i\). Applying the projection map \(\pi_i\) to this tuple, we have: \(\pi_i(\left(\alpha_{1,0}, \dots, \alpha_{i-1,0}, \alpha_i, \alpha_{i+1,0}, \dots, \alpha_{k,0}\right)) = \alpha_i\). So for every \(\alpha_i \in M_i\), we have found the tuple that maps to \(\alpha_i\) under \(\pi_i\). Thus, \(\pi_i\) is a surjective (onto) function. Since we have shown that for each \(i\), the projection map \(\pi_i: M_1 \times \dots \times M_k \rightarrow M_i\) is an \(R\)-linear map and surjective, we have completed the proof.

Key concepts

Surjective Function

Understanding the concept of a surjective function is crucial in various branches of mathematics, including algebra. A surjective function, commonly known as an 'onto' function, is a type of mapping where every element in the target set has a preimage in the domain. In formal terms, for a function to be surjective, every element b in the codomain B must be the image of at least one element a in the domain A, such that f(a) = b.

In the context of the exercise, the projection map \(\pi_i\) demonstrates surjectivity by mapping each element of the set \(M_i\) to a corresponding tuple in the product of all modules. This ensures that no element in the codomain \(M_i\) is left without a preimage, thus fulfilling the criterion for a function to be surjective. Clear understanding of surjective functions is essential for students as they underpin many concepts in advanced mathematics, including module theory and the study of algebraic structures.

Module Theory

Module theory is an extension of ring theory, a fundamental part of abstract algebra. Modules can be seen as generalizations of vector spaces, where instead of a field, we have a ring R. An R-module is a set equipped with an addition operation and an action of R satisfying certain axioms akin to those of vector spaces.

For a deeper understanding, consider the properties of R-modules. They include closure under addition, existence of an additive identity, existence of additive inverses, compatibility of scalar multiplication with field multiplication, and distributivity of scalar multiplication over both field addition and vector addition.

In the given exercise, the entities \(M_1, \ldots, M_k\) are all R-modules, making the collective product \(M_1 \times \cdots \times M_k\) an R-module as well. This is crucial in solving the exercise, as understanding that each individual projection \(\pi_i\) is an R-linear map that reflects the module structure is vital in proving the surjectivity of these maps.

Algebraic Structures

Algebraic structures form the backbone of abstract algebra and are defined by a set accompanied by one or more operations that satisfy a set of axioms. Examples of such structures include groups, rings, fields, and modules. These structures help us understand the properties of sets that are not only under the influence of arithmetical operations but also more complex interactions dictated by their respective axioms.

For instance, in module theory, the set along with addition and scalar multiplication operations forms a module, an algebraic structure when the set and operations satisfy certain axioms. Grasping the fundamental properties of algebraic structures allows students to navigate through more advanced topics and exercises, such as the one in question, which involves the interplay of multiple modules, another kind of algebraic structure, and the algebraic operations defined on them.