Rings form the backbone of noncommutative geometry, generalizing familiar number systems and algebraic structures. They combine addition and multiplication operations, providing a unified framework for studying integers, polynomials, matrices, and functions.
Rings can be commutative or noncommutative, with the latter being particularly important in noncommutative geometry. Examples include matrix rings and quaternions, which capture the essence of noncommutative spaces and their unique properties.
Definition of rings
Rings are algebraic structures that generalize the arithmetic operations of addition and multiplication
They provide a unified framework for studying various mathematical objects, such as integers, polynomials, matrices, and functions
Rings play a central role in noncommutative geometry, where they are used to describe the geometry of noncommutative spaces
Additive group structure
Top images from around the web for Additive group structure
MultiwayGroup | Wolfram Function Repository View original
Is this image relevant?
visualization - Viewing an abelian group using cayley diagram - Mathematics Stack Exchange View original
Is this image relevant?
MultiwayGroup | Wolfram Function Repository View original
Is this image relevant?
MultiwayGroup | Wolfram Function Repository View original
Is this image relevant?
visualization - Viewing an abelian group using cayley diagram - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
Top images from around the web for Additive group structure
MultiwayGroup | Wolfram Function Repository View original
Is this image relevant?
visualization - Viewing an abelian group using cayley diagram - Mathematics Stack Exchange View original
Is this image relevant?
MultiwayGroup | Wolfram Function Repository View original
Is this image relevant?
MultiwayGroup | Wolfram Function Repository View original
Is this image relevant?
visualization - Viewing an abelian group using cayley diagram - Mathematics Stack Exchange View original
Is this image relevant?
1 of 3
A ring (R,+,⋅) has an underlying additive abelian group structure (R,+)
The addition operation + is associative, commutative, and has an identity element (usually denoted by 0)
Each element a∈R has an additive inverse −a such that a+(−a)=0
Multiplicative monoid structure
The multiplication operation ⋅ in a ring is associative, i.e., (a⋅b)⋅c=a⋅(b⋅c) for all a,b,c∈R
The multiplication may or may not be commutative
A ring may or may not have a multiplicative identity element (usually denoted by 1)
Distributive property
The multiplication in a ring is distributive over addition, i.e., a⋅(b+c)=(a⋅b)+(a⋅c) and (a+b)⋅c=(a⋅c)+(b⋅c) for all a,b,c∈R
This property connects the additive and multiplicative structures of the ring
Commutative vs noncommutative rings
A ring is called commutative if its multiplication is commutative, i.e., a⋅b=b⋅a for all a,b∈R
Examples of commutative rings include the integers Z, real numbers R, and polynomial rings R[x]
Noncommutative rings, where multiplication is not always commutative, are of particular interest in noncommutative geometry
Matrix rings Mn(R) and quaternion rings H are examples of noncommutative rings
Unital vs non-unital rings
A ring is called unital if it has a multiplicative identity element 1 such that 1⋅a=a⋅1=a for all a∈R
Most commonly studied rings, such as Z, R, and Mn(R), are unital
Non-unital rings, which lack a multiplicative identity, also appear in some contexts
An example of a non-unital ring is the ring of even integers 2Z under the usual addition and multiplication
Examples of rings
Rings provide a unified language for studying various mathematical objects and their algebraic properties
Many familiar number systems and algebraic structures can be viewed as rings
Noncommutative geometry often focuses on noncommutative rings, which capture the essence of noncommutative spaces
Integer rings
The set of integers Z forms a commutative unital ring under the usual addition and multiplication
Z is the prototypical example of a ring and serves as a starting point for studying more general rings
Many ring-theoretic properties, such as primality and factorization, are first studied in the context of Z
Polynomial rings
Given a ring R, the set of polynomials with coefficients in R forms a ring R[x] under the usual addition and multiplication of polynomials
If R is commutative, then R[x] is also commutative; however, R[x] is noncommutative when R is noncommutative
Polynomial rings are crucial in algebraic geometry and serve as a bridge between commutative and noncommutative rings
Matrix rings
The set of n×n matrices with entries in a ring R forms a ring Mn(R) under matrix addition and multiplication
Matrix rings are inherently noncommutative for n≥2, as matrix multiplication is not commutative in general
Mn(R) is a central example of a and plays a significant role in the study of representations of rings and modules
Quaternion rings
The quaternions H form a noncommutative , extending the complex numbers
Quaternions are generated by four basis elements 1,i,j,k with the multiplication rules i2=j2=k2=ijk=−1
Quaternions have applications in various fields, including geometry, physics, and computer graphics
Group rings
Given a ring R and a group G, the group ring R[G] is a ring constructed by forming formal linear combinations of elements of G with coefficients in R
The multiplication in R[G] is defined by extending the group multiplication linearly
Group rings provide a way to study the interplay between ring theory and group theory
When G is a noncommutative group, R[G] is a noncommutative ring, even if R is commutative
Ring homomorphisms
Ring homomorphisms are structure-preserving maps between rings that respect both the additive and multiplicative operations
They allow us to study the relationships between different rings and transfer properties from one ring to another
Ring homomorphisms are essential tools in the study of ring theory and noncommutative geometry
Definition and properties
A ring homomorphism ϕ:R→S is a function between rings R and S satisfying:
ϕ(a+b)=ϕ(a)+ϕ(b) for all a,b∈R (additive homomorphism)
ϕ(a⋅b)=ϕ(a)⋅ϕ(b) for all a,b∈R (multiplicative homomorphism)
If R and S are unital rings, then ϕ(1R)=1S, where 1R and 1S are the multiplicative identities of R and S, respectively
The composition of two ring homomorphisms is again a ring homomorphism
Kernel and image of homomorphisms
The kernel of a ring homomorphism ϕ:R→S is the set ker(ϕ)={a∈R:ϕ(a)=0S}, where 0S is the additive identity of S
ker(ϕ) is an of R, and it measures the "degree of injectivity" of ϕ
The image of ϕ is the set im(ϕ)={ϕ(a):a∈R}, which is a subring of S
The image measures the "degree of surjectivity" of ϕ
Isomorphisms and automorphisms
A ring homomorphism ϕ:R→S is called an isomorphism if it is bijective (i.e., injective and surjective)
Two rings R and S are isomorphic, denoted by R≅S, if there exists an isomorphism between them
An automorphism is an isomorphism from a ring to itself, i.e., a bijective homomorphism ϕ:R→R
The set of all automorphisms of a ring R forms a group under composition, denoted by Aut(R)
Endomorphism rings
An endomorphism of a ring R is a homomorphism from R to itself, i.e., a function ϕ:R→R that is both additive and multiplicative
The set of all endomorphisms of R forms a ring under pointwise addition and composition, called the endomorphism ring End(R)
End(R) is a unital ring, with the identity endomorphism serving as the multiplicative identity
Studying the endomorphism ring of a given ring can reveal important structural properties of the ring
Ideals and quotient rings
Ideals are special subsets of a ring that absorb multiplication and allow for the construction of quotient rings
They play a fundamental role in the study of ring theory and provide a way to create new rings from existing ones
Quotient rings generalize the notion of modular arithmetic and are essential in noncommutative geometry
Definition of ideals
A subset I of a ring R is called a left ideal if:
(I,+) is a subgroup of (R,+)
For all a∈R and x∈I, we have a⋅x∈I
Similarly, a right ideal is a subset I satisfying the above conditions with the second condition replaced by x⋅a∈I for all a∈R and x∈I
A two-sided ideal (or simply an ideal) is a subset that is both a left and a right ideal
Left, right, and two-sided ideals
The distinction between left, right, and two-sided ideals is important in noncommutative rings
In a , every left or right ideal is automatically a two-sided ideal
Examples of left and right ideals that are not two-sided can be found in matrix rings and other noncommutative rings
The sum and intersection of two left (resp. right) ideals is again a left (resp. right) ideal
Principal and finitely generated ideals
A left (resp. right) ideal I is called principal if it can be generated by a single element, i.e., I=Ra (resp. I=aR) for some a∈R
More generally, a left (resp. right) ideal is finitely generated if it can be generated by a finite set of elements
In commutative rings, principal ideals play a crucial role in the study of unique factorization and prime ideals
Finitely generated ideals are important in the context of Noetherian rings
Quotient rings and isomorphism theorems
Given a two-sided ideal I of a ring R, the R/I is the set of cosets a+I={a+x:x∈I} with addition and multiplication defined as (a+I)+(b+I)=(a+b)+I and (a+I)(b+I)=(ab)+I
The quotient ring R/I is a well-defined ring, and the natural projection π:R→R/I given by π(a)=a+I is a surjective ring homomorphism with kernel I
The first isomorphism theorem states that if ϕ:R→S is a ring homomorphism, then R/ker(ϕ)≅im(ϕ)
The second and third isomorphism theorems relate quotients of quotient rings and quotients by sums and intersections of ideals
Prime and maximal ideals
An ideal P of a commutative ring R is called prime if P=R and whenever ab∈P for a,b∈R, then either a∈P or b∈P
An ideal M of a ring R is called maximal if M=R and there is no ideal I satisfying M⊊I⊊R
In a commutative ring, every maximal ideal is prime, but the converse is not always true
The spectrum of a commutative ring R, denoted by Spec(R), is the set of all prime ideals of R and is a central object of study in algebraic geometry
Modules over rings
Modules are a generalization of vector spaces, where the scalars are elements of a ring instead of a field
They provide a unified framework for studying various algebraic structures, such as abelian groups, vector spaces, and ideals
Modules are essential in noncommutative geometry, as they allow for the development of homological and cohomological methods
Definition and examples of modules
Given a ring R, a left R-module is an abelian group (M,+) together with a scalar multiplication R×M→M, denoted by (r,m)↦rm, satisfying:
(r+s)m=rm+sm for all r,s∈R and m∈M
r(m+n)=rm+rn for all r∈R and m,n∈M
(rs)m=r(sm) for all r,s∈R and m∈M
1m=m for all m∈M, if R is unital
Examples of modules include vector spaces over fields, abelian groups (as Z-modules), and ideals of a ring (as modules over that ring)
Submodules and quotient modules
A submodule of an R-module M is a subgroup N of M that is closed under scalar multiplication, i.e., rn∈N for all r∈R and n∈N
Given a submodule N of M, the quotient module M/N is the set of cosets m+N={m+n:n∈N} with addition and scalar multiplication defined as (m+N)+(m′+N)=(m+m′)+N and r(m+N)=rm+N
The quotient module M/N is a well-defined R-module, and the natural projection π:M→M/N given by π(m)=m+N is a surjective R-module homomorphism with kernel N
Free modules and bases
An R-module F is called free if it has a basis, i.e., a subset B⊆F such that every element of F can be uniquely expressed as a finite linear combination of elements from B with coefficients in R
The cardinality of a basis is called the rank of the
Examples of free modules include vector spaces over fields and the R-module Rn for any n∈N
Every module is a quotient of a free module, which allows for the study of modules using free resolutions
Tensor products of modules
Given two R-modules M and N, the tensor product M⊗RN is an R-module that represents the "most general bilinear operation" on M and N
The tensor product is constructed as the quotient of the free R-module generated by the set M×N by the submodule generated by elements of the form (m+m′,n)−(m,n)−(m′,n), (m,n+n′)−(m,n)−(m,n′), and (rm,n)−(m,rn) for m,m′∈M, n,n′∈N, and r∈R
Tensor products have numerous applications in algebra, geometry, and topology, such as the study of bilinear forms, the construction of the algebra of differential forms, and the definition of homology and cohomology theories
Projective and injective modules
An R-module P is called projective if for every surjective R-module homomorphism f:M→N and every R-module homomorphism g:P→N, there exists an R-module homomorphism h:P→M such that f∘h=g
Projective modules are direct summands of free modules and play a crucial role in homological algebra
An R-module I is called injective if for every injective R-module homomorphism f:M→N and every R-module homomorphism g:M→I, there exists an R-module homomorphism h:N→I such that h∘f=g
Injective modules are important in the study of cohomology and the construction of injective
Key Terms to Review (18)
Associativity: Associativity is a fundamental property of certain binary operations that states the way in which operations are grouped does not affect the outcome. Specifically, for any three elements, the equation $(a * b) * c = a * (b * c)$ holds true for an associative operation. This property is crucial in various mathematical structures and allows for simplification of expressions and calculations.
Chinese Remainder Theorem: The Chinese Remainder Theorem is a result in number theory that provides a way to solve systems of simultaneous congruences with different moduli. It states that if the moduli are pairwise coprime, then there exists a unique solution modulo the product of the moduli. This theorem is crucial in the study of rings and modular arithmetic, as it allows for simplifying calculations by breaking them down into smaller, more manageable components.
Commutative Ring: A commutative ring is an algebraic structure consisting of a set equipped with two binary operations: addition and multiplication, where the set is closed under both operations, addition is associative and commutative, multiplication is associative, and multiplication distributes over addition. This structure is fundamental in abstract algebra and lays the groundwork for further studies in algebraic systems and number theory.
Distributivity: Distributivity is a fundamental property in algebra that describes how multiplication interacts with addition. Specifically, it states that for any numbers (or elements) a, b, and c, the equation $$a \cdot (b + c) = a \cdot b + a \cdot c$$ holds true. This property is crucial in the context of rings, as it ensures that operations within the ring are coherent and can be manipulated in a predictable manner, allowing for the simplification of expressions and equations.
Division Ring: A division ring is a type of algebraic structure that consists of a set equipped with two binary operations, addition and multiplication, where every non-zero element has a multiplicative inverse. Unlike fields, division rings do not require multiplication to be commutative. This means that while division rings share many properties with fields, they allow for more flexibility in the arrangement of elements, particularly concerning multiplication.
Free Module: A free module is a type of module over a ring that has a basis, meaning it can be expressed as a direct sum of copies of the ring. This property makes free modules very similar to vector spaces, where the elements of the module can be represented as linear combinations of the basis elements. The significance of free modules lies in their structure, which allows them to have properties that facilitate computations and theoretical developments in algebraic structures involving rings and modules.
Ideal: An ideal is a special subset of a ring that allows for the construction of quotient rings and provides a framework for factoring elements within that ring. Ideals help in understanding the structure of rings, as they define a way to 'collapse' the ring into simpler components, thus linking various algebraic concepts together. They play a key role in algebraic structures, paving the way for concepts such as homomorphisms, modules, and representations.
Injective Homomorphism: An injective homomorphism is a type of function between two algebraic structures, such as rings, that preserves the operations and maps distinct elements from the first structure to distinct elements in the second. This means that if two elements are different in the original structure, their images under the homomorphism will also be different, ensuring that no information is lost in the mapping. In the context of rings, injective homomorphisms help in understanding how one ring can be embedded into another, maintaining its structure.
Matrix Ring: A matrix ring is a set of matrices of a given size with entries from a specific ring, where the operations of addition and multiplication are defined as usual for matrices. This structure forms a ring itself, meaning it satisfies the properties of closure, associativity, distributivity, and has an additive identity. Matrix rings are important in various mathematical areas, including linear algebra and noncommutative geometry, as they allow for the study of linear transformations and their properties in a systematic way.
Noncommutative Ring: A noncommutative ring is a type of algebraic structure where the multiplication operation does not satisfy the commutative property, meaning that for some elements a and b in the ring, it holds that a*b ≠ b*a. This property allows for a richer structure that can model various mathematical and physical phenomena, particularly in areas such as quantum mechanics and noncommutative geometry. Noncommutative rings contrast with commutative rings, where the order of multiplication does not affect the outcome.
Polynomial Ring: A polynomial ring is a mathematical structure consisting of polynomials, which are expressions formed by adding and multiplying variables raised to non-negative integer powers, along with coefficients from a specified ring. It serves as a fundamental concept in algebra, allowing for the extension of ring operations to include polynomial expressions, enabling the study of algebraic structures and equations.
Projective Module: A projective module is a type of module that has the lifting property, meaning it can be thought of as a 'generalized' vector space over a ring. These modules can be expressed as direct summands of free modules, making them crucial in the study of homological algebra. Their properties relate closely to rings, modules, topological algebras, and various concepts in noncommutative geometry.
Quotient Ring: A quotient ring is a type of ring formed by partitioning a given ring into equivalence classes using a two-sided ideal. It captures the notion of 'dividing' out by the ideal, allowing for a simpler structure that retains key properties of the original ring. By taking this approach, quotient rings facilitate the study of rings by enabling operations in a more manageable setting.
Ring addition: Ring addition refers to the binary operation of addition defined in the context of a ring, a mathematical structure that consists of a set equipped with two operations: addition and multiplication. In a ring, the addition operation must satisfy specific properties, such as closure, associativity, the existence of an additive identity (usually denoted as 0), and the existence of additive inverses for every element in the ring. This foundational operation is crucial for exploring the more complex interactions within the structure of rings.
Ring Isomorphism Theorem: The Ring Isomorphism Theorem states that if two rings are isomorphic, there exists a bijective ring homomorphism between them that preserves both the addition and multiplication operations. This theorem highlights the idea that isomorphic rings, while they may appear different in structure or representation, are fundamentally the same in terms of their algebraic properties.
Ring multiplication: Ring multiplication is an operation in ring theory that combines two elements from a ring to produce a third element, following specific rules. This operation must be associative and distributive over addition, and it plays a crucial role in defining the structure of rings, which are algebraic systems with two operations: addition and multiplication.
Ring with unity: A ring with unity is a type of ring that includes a multiplicative identity element, usually denoted as 1, such that for any element 'a' in the ring, multiplying 'a' by 1 yields 'a'. This concept is vital because it allows for the definition of multiplicative inverses and facilitates various algebraic structures that arise in ring theory, enhancing the analysis of rings and their properties.
Surjective Homomorphism: A surjective homomorphism is a function between two algebraic structures, such as rings, that preserves the structure and maps every element from the first structure onto the second, ensuring that every element in the second structure is covered. This type of mapping is crucial because it guarantees that the image of the homomorphism is equal to the entire codomain, which affects properties like isomorphism and quotient structures in ring theory.