1.4 Modules

Modules generalize vector spaces by allowing scalars from rings instead of fields. They're crucial in noncommutative geometry, providing a framework for studying spaces with noncommutative coordinate rings. Understanding modules is key to working with noncommutative spaces.

Modules combine an abelian group structure with scalar multiplication from a ring. This allows for more flexible algebraic structures than vector spaces. The notes cover various types of modules, submodules, quotients, homomorphisms, and constructions like direct sums and tensor products.

Definition of modules

• Modules are a fundamental concept in algebra that generalize the notion of vector spaces by allowing scalars to come from a ring instead of a field
• Modules play a crucial role in noncommutative geometry as they provide a framework for studying spaces with noncommutative coordinate rings
• Understanding the properties and structure of modules is essential for working with noncommutative spaces and their associated algebraic objects

Modules over rings

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• A module over a ring $R$ is an abelian group $M$ together with a scalar multiplication map $R \times M \to M$ that satisfies certain compatibility conditions
• The scalar multiplication map associates an element of the ring $R$ and an element of the module $M$ to produce another element of $M$
• The compatibility conditions ensure that the scalar multiplication interacts well with the ring and abelian group structures

Abelian group structure

• A module $M$ is an abelian group under addition, meaning that elements of $M$ can be added together and the addition operation satisfies the following properties:
  • Associativity: $(a + b) + c = a + (b + c)$ for all $a, b, c \in M$
  • Commutativity: $a + b = b + a$ for all $a, b \in M$
  • Identity element: There exists an element $0 \in M$ such that $a + 0 = a$ for all $a \in M$
  • Inverse elements: For each $a \in M$, there exists an element $-a \in M$ such that $a + (-a) = 0$

Scalar multiplication

• The scalar multiplication map $R \times M \to M$ satisfies the following properties for all $r, s \in R$ and $a, b \in M$:
  • $(rs)a = r(sa)$
  • $(r + s)a = ra + sa$
  • $r(a + b) = ra + rb$
  • $1a = a$, where $1$ is the multiplicative identity of the ring $R$
• These properties ensure that the scalar multiplication is compatible with the ring structure and distributes over the abelian group structure of the module

Types of modules

• There are several important types of modules that arise in various contexts, each with its own set of properties and characteristics
• Understanding the different types of modules is crucial for studying the structure and behavior of modules in noncommutative geometry

Free modules

• A module $M$ over a ring $R$ is called a free module if it has a basis, i.e., a subset $B \subseteq M$ such that every element of $M$ can be uniquely expressed as a finite linear combination of elements from $B$ with coefficients in $R$
• Free modules are analogous to vector spaces in linear algebra and serve as building blocks for more general modules
• Examples of free modules include:
  • The set of polynomials $R[x]$ over a ring $R$, with basis $\{1, x, x^2, \ldots\}$
  • The set of $n$-tuples $R^n$ over a ring $R$, with basis $\{e_1, \ldots, e_n\}$, where $e_i$ has a $1$ in the $i$-th position and $0$s elsewhere

Finitely generated modules

• A module $M$ over a ring $R$ is called finitely generated if there exists a finite subset $\{x_1, \ldots, x_n\} \subseteq M$ such that every element of $M$ can be expressed as a linear combination of $x_1, \ldots, x_n$ with coefficients in $R$
• Finitely generated modules are a generalization of finite-dimensional vector spaces and play a significant role in the study of modules
• Examples of finitely generated modules include:
  • Any finite abelian group, considered as a module over the integers
  • The quotient module $R^n/U$, where $U$ is a submodule of $R^n$ generated by a finite set of elements

Projective modules

• A module $P$ over a ring $R$ is called projective if it satisfies the following property: for any surjective module homomorphism $f: M \to N$ and any module homomorphism $g: P \to N$, there exists a module homomorphism $h: P \to M$ such that $f \circ h = g$
• Projective modules can be thought of as direct summands of free modules and have important applications in homological algebra and K-theory
• Examples of projective modules include:
  • Free modules
  • The direct sum of projective modules

Injective modules

• A module $I$ over a ring $R$ is called injective if it satisfies the following property: for any injective module homomorphism $f: M \to N$ and any module homomorphism $g: M \to I$, there exists a module homomorphism $h: N \to I$ such that $h \circ f = g$
• Injective modules are dual to projective modules and play a crucial role in the study of homological algebra
• Examples of injective modules include:
  • The abelian group $\mathbb{Q}/\mathbb{Z}$, considered as a module over the integers
  • The abelian group of $p$-adic integers, considered as a module over the integers

Simple modules

• A module $M$ over a ring $R$ is called simple if it is nonzero and has no submodules other than $\{0\}$ and $M$ itself
• Simple modules are the building blocks of more complex modules and are analogous to irreducible representations in representation theory
• Examples of simple modules include:
  • The abelian group $\mathbb{Z}/p\mathbb{Z}$, where $p$ is a prime, considered as a module over itself
  • The abelian group $\mathbb{C}$, considered as a module over the complex numbers

Submodules and quotient modules

• Submodules and quotient modules are fundamental constructions in the study of modules that allow us to understand the internal structure of modules and their relationships to one another
• These concepts are essential for developing the theory of modules and their applications in noncommutative geometry

Submodule definition

• A submodule of a module $M$ over a ring $R$ is a subset $N \subseteq M$ that is itself a module over $R$ under the same addition and scalar multiplication operations as $M$
• In other words, a submodule is a nonempty subset of $M$ that is closed under addition and scalar multiplication
• Examples of submodules include:
  • The trivial submodules $\{0\}$ and $M$ itself
  • The kernel of a module homomorphism $f: M \to N$, defined as $\ker(f) = \{x \in M : f(x) = 0\}$

Submodule properties

• Submodules satisfy several important properties that are analogous to those of subspaces in linear algebra:
  • The intersection of any collection of submodules of $M$ is again a submodule of $M$
  • The sum of two submodules $N_1$ and $N_2$ of $M$, defined as $N_1 + N_2 = \{x_1 + x_2 : x_1 \in N_1, x_2 \in N_2\}$, is a submodule of $M$
  • If $N$ is a submodule of $M$ and $I$ is an ideal of the ring $R$, then $IN = \{rx : r \in I, x \in N\}$ is a submodule of $M$

Quotient modules

• Given a module $M$ over a ring $R$ and a submodule $N$ of $M$, the quotient module $M/N$ is defined as the set of equivalence classes $[x] = \{x + y : y \in N\}$ for $x \in M$, with addition and scalar multiplication given by $[x] + [y] = [x + y]$ and $r[x] = [rx]$ for $r \in R$
• The quotient module $M/N$ can be thought of as the result of "collapsing" the submodule $N$ to zero and considering the resulting structure on the equivalence classes
• The quotient module construction is a powerful tool for studying the structure of modules and their relationships to one another

Isomorphism theorems for modules

• The isomorphism theorems for modules are a set of fundamental results that describe the relationship between submodules, quotient modules, and module homomorphisms
• The first isomorphism theorem states that if $f: M \to N$ is a module homomorphism, then $M/\ker(f) \cong \operatorname{im}(f)$, where $\operatorname{im}(f) = \{f(x) : x \in M\}$ is the image of $f$
• The second isomorphism theorem states that if $N_1$ and $N_2$ are submodules of $M$ with $N_1 \subseteq N_2$, then $(M/N_1)/(N_2/N_1) \cong M/N_2$
• The third isomorphism theorem states that if $N$ and $K$ are submodules of $M$ with $K \subseteq N$, then $(M/K)/(N/K) \cong M/N$
• These isomorphism theorems provide a powerful framework for understanding the relationships between modules and their substructures

Module homomorphisms

• Module homomorphisms are structure-preserving maps between modules that play a fundamental role in the study of modules and their relationships to one another
• Understanding the properties and behavior of module homomorphisms is essential for developing the theory of modules and their applications in noncommutative geometry

Definition of module homomorphisms

• A module homomorphism between two modules $M$ and $N$ over a ring $R$ is a function $f: M \to N$ that satisfies the following properties for all $x, y \in M$ and $r \in R$:
  • $f(x + y) = f(x) + f(y)$
  • $f(rx) = rf(x)$
• In other words, a module homomorphism is a map that preserves the module structure, i.e., it respects addition and scalar multiplication
• Examples of module homomorphisms include:
  • The zero homomorphism $f: M \to N$ defined by $f(x) = 0$ for all $x \in M$
  • The identity homomorphism $\operatorname{id}_M: M \to M$ defined by $\operatorname{id}_M(x) = x$ for all $x \in M$

Kernel and image of homomorphisms

• The kernel of a module homomorphism $f: M \to N$ is the submodule of $M$ defined as $\ker(f) = \{x \in M : f(x) = 0\}$
• The kernel consists of all elements of $M$ that are mapped to the zero element of $N$ under the homomorphism $f$
• The image of a module homomorphism $f: M \to N$ is the submodule of $N$ defined as $\operatorname{im}(f) = \{f(x) : x \in M\}$
• The image consists of all elements of $N$ that are "reachable" from elements of $M$ under the homomorphism $f$
• The kernel and image of a module homomorphism provide important information about the structure of the homomorphism and the relationship between the modules involved

Isomorphisms of modules

• Two modules $M$ and $N$ over a ring $R$ are said to be isomorphic if there exists a module homomorphism $f: M \to N$ that is both injective (one-to-one) and surjective (onto)
• An isomorphism of modules is a bijective module homomorphism, and we write $M \cong N$ to denote that $M$ and $N$ are isomorphic
• Isomorphic modules have the same underlying structure and can be thought of as essentially the same object, up to relabeling of elements
• Examples of module isomorphisms include:
  • The modules $\mathbb{R}^n$ and $\mathbb{R}^m$ are isomorphic if and only if $n = m$
  • The modules $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$ are isomorphic if and only if $n = m$

Direct sums and products

• Direct sums and direct products are constructions that allow us to combine modules in a way that preserves their individual structures
• These constructions are important for understanding the decomposition of modules into simpler components and for studying the behavior of modules under various operations

Direct sum of modules

• The direct sum of a family of modules $\{M_i\}_{i \in I}$ over a ring $R$, denoted by $\bigoplus_{i \in I} M_i$, is the module consisting of all tuples $(x_i)_{i \in I}$ with $x_i \in M_i$ for each $i \in I$ and $x_i = 0$ for all but finitely many $i \in I$
• Addition and scalar multiplication in the direct sum are defined componentwise:
  • $(x_i)_{i \in I} + (y_i)_{i \in I} = (x_i + y_i)_{i \in I}$
  • $r(x_i)_{i \in I} = (rx_i)_{i \in I}$ for $r \in R$
• The direct sum can be thought of as the "smallest" module containing copies of each $M_i$ as submodules, with no additional relations between elements of different $M_i$

Direct product of modules

• The direct product of a family of modules $\{M_i\}_{i \in I}$ over a ring $R$, denoted by $\prod_{i \in I} M_i$, is the module consisting of all tuples $(x_i)_{i \in I}$ with $x_i \in M_i$ for each $i \in I$
• Addition and scalar multiplication in the direct product are defined componentwise, just as in the direct sum
• The direct product can be thought of as the "largest" module containing copies of each $M_i$ as submodules, allowing for arbitrary tuples of elements from the $M_i$

Properties of direct sums and products

• Direct sums and direct products satisfy several important properties that make them useful in the study of modules:
  • The direct sum $\bigoplus_{i \in I} M_i$ is isomorphic to the direct product $\prod_{i \in I} M_i$ if and only if $I$ is a finite set
  • The direct sum and direct product are both associative and commutative up to isomorphism
  • If each $M_i$ is a submodule of a module $M$, then the direct sum $\bigoplus_{i \in I} M_i$ is a submodule of $M$ if and only if the sum of the $M_i$ is direct (i.e., $M_i \cap \sum_{j \neq i} M_j = \{0\}$ for all $i \in I$)
• Understanding the properties of direct sums and direct products is essential for studying the decomposition and structure of modules in various contexts

Tensor products of modules

• The tensor product is a fundamental construction in algebra that allows us to combine modules in a way that captures bilinear relationships between their elements
• Tensor products play a crucial role in many areas of mathematics, including algebraic geometry, representation theory, and homological algebra, and are essential for studying modules in noncommutative settings

Definition of tensor product

• Given modules $M$ and $N$ over a ring $R$, the tensor product of $M$ and $N$, denoted by $M \otimes_R N$, is the module generated by symbols of the form $x \otimes y$ for $x \in M$ and $y \in N$, subject to the following relations for all $x, x' \in M$, $y, y' \in N$, and $r \in R$:
  • $(x + x') \otimes y = x \otimes y + x' \otimes y$
  • $x \otimes (y + y') = x \otimes y + x \otimes y'$
  • $(rx) \otimes y = x \otimes (ry) = r(x \otimes y)$
• The tensor product can be thought of as a way to "multiply" elements of $M$ and $N$ while respecting the module structures and capturing bilinear relationships

Universal property of tensor products

• The tensor product satisfies a universal property that characterizes it uniquely up to isomorphism:
  • Given modules $M