Question

Show that if \(M=M_{1} \times M_{2}\) for \(R\) -modules \(M_{1}\) and \(M_{2},\) and \(N_{1}\) is a subgroup of \(M_{1}\) and \(N_{2}\) is a subgroup of \(M_{2},\) then we have an \(R\) -module isomorphism \(M /\left(N_{1} \times N_{2}\right) \cong M_{1} / N_{1} \times M_{2} / N_{2}\)

Short answer

Question: Prove that there exists an R-module isomorphism between the quotient module M/(N1 x N2) and the direct product of the quotient modules M1/N1 x M2/N2.

Step-by-step solution

Define the mapping

Define a mapping φ: M → M1/N1 x M2/N2 by: $$ \phi((m_{1}, m_{2})) = (m_{1}+N_{1}, m_{2}+N_{2}) $$ for all \((m_{1}, m_{2}) \in M\).

Prove φ is a homomorphism

We now need to show that φ is a homomorphism. Let \((m_{1}, m_{2}), (n_{1}, n_{2}) \in M\) and \(r \in R\). Then: $$ \phi((m_{1}, m_{2})+(n_{1}, n_{2})) = \phi((m_{1}+n_{1}, m_{2}+n_{2})) = (m_{1}+n_{1}+N_{1}, m_{2}+n_{2}+N_{2}) $$ and $$ \phi((m_{1}, m_{2})) + \phi((n_{1},n_{2})) = (m_{1}+N_{1}, m_{2}+N_{2}) + (n_{1}+N_{1},n_{2}+N_{2}) = (m_{1}+n_{1}+N_{1}, m_{2}+n_{2}+N_{2}) $$ Since the two expressions are equal, φ is a homomorphism.

Prove φ is R-linear

Next, we need to show that φ is R-linear. Let \((m_{1}, m_{2}) \in M\) and \(r \in R\). Then: $$ \phi(r(m_{1},m_{2})) = \phi((rm_{1},rm_{2})) = (rm_{1}+N_{1},rm_{2}+N_{2}) $$ and $$ r\phi((m_{1},m_{2})) = r(m_{1}+N_{1},m_{2}+N_{2}) = (rm_{1}+N_{1},rm_{2}+N_{2}) $$ Since the two expressions are equal, φ is R-linear.

Find the kernel of φ

We now need to find the kernel of φ. The kernel of φ is: $$ \text{ker}\, \phi = \{(m_{1},m_{2}) \in M \mid \phi((m_{1},m_{2}))=(N_{1},N_{2})\} = \{(m_{1},m_{2}) \in M \mid m_{1} \in N_{1} \text{ and } m_{2} \in N_{2}\} = N_{1}\times N_{2} $$

Apply the First Isomorphism Theorem

By the First Isomorphism Theorem for R-modules, we have: $$ M / \text{ker}\, \phi \cong \text{Im}\, \phi $$ Since the kernel of φ is N1 x N2, we have: $$ M / (N_{1} \times N_{2}) \cong \text{Im}\, \phi $$

Prove that the Image of φ is M1/N1 x M2/N2

We need to show that the image of φ is M1/N1 x M2/N2. Let \((a_{1} + N_{1}, a_{2} + N_{2}) \in M1/N1 \times M2/N2\). Then, we have: $$ \phi((a_{1}, a_{2})) = (a_{1} + N_{1}, a_{2} + N_{2}) $$ Hence, the image of φ is M1/N1 x M2/N2.

Conclude that M/(N1 x N2) is isomorphic to M1/N1 x M2/N2

Since we have shown that there exists an R-module isomorphism between the quotient module M/(N1 x N2) and the direct product of the quotient modules M1/N1 x M2/N2, we can conclude: $$ M / (N_{1} \times N_{2}) \cong M_{1} / N_{1} \times M_{2} / N_{2} $$

Key concepts

Module Homomorphism

Understanding module homomorphisms is crucial when dealing with module theory in algebra. A module homomorphism is a function that preserves the module structure between two given modules. Specifically, for modules M and N over the same ring R, a function φ: M → N is a homomorphism if it satisfies two main conditions.

First, it must be additive; that is, for any elements m₁, m₂ in M, φ(m₁ + m₂) = φ(m₁) + φ(m₂). Second, it has to respect scalar multiplication from the ring R; meaning that for any r in R and m in M, we have φ(rm) = rφ(m).

When a homomorphism is injective, surjective, or bijective, we use special names—monomorphism, epimorphism, and isomorphism, respectively. The presence of an isomorphism between two modules indicates that they are structurally identical or 'the same' in terms of module theory.

R-linear Mapping

Often used interchangeably with module homomorphism, an R-linear mapping is a specific case where the modules are considered as vector spaces over a field R, and the linear transformations between them maintain the structure of these spaces. In general, for any ring R, a map between two R-modules that preserves the operations of addition and scalar multiplication—as per the definitions provided for module homomorphism—is called R-linear.

An R-linear mapping's key traits include its kernel and image. The kernel is the set of elements in the domain that map to the zero element in the codomain, and the image is the set of elements in the codomain that have a preimage in the domain. These concepts play a vital role in understanding the structure and behavior of modules through mapping.

First Isomorphism Theorem

The First Isomorphism Theorem is a foundational result in algebra that applies to various algebraic structures, including groups, rings, and modules. Pertaining to modules, it states that if φ: M → N is a module homomorphism, then the quotient module of M by the kernel of φ is isomorphic to the image of φ. This can be formally written as M / ker(φ) ≅ Im(φ).

The theorem not only confirms that the kernel and image of a homomorphism determine each other up to isomorphism, but it also enables simplification of complex structures by breaking them down into related, more manageable parts. As such, it serves as a bridge connecting the concepts of kernel, image, and quotient structures within the universe of modules.

Quotient Module

In the study of modules, the idea of a quotient module comes into play when we wish to 'mod out' a submodule from a given module. For a module M and a submodule N, the quotient module M/N is comprised of cosets of N in M. A coset is formed by adding a fixed element of M to every element of N.

The quotient module carries a natural module structure inherited from M, and enables mathematicians to study properties of M relative to N. It can provide significant insights into the composition of the original module, particularly when combined with concepts such as homomorphisms and isomorphism theorems. The formation of quotient modules is a central operation in many areas of algebra and has powerful implications for simplifying complex algebraic problems.