In ring theory

Ring theory

In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...

 or abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, a ring homomorphism is a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 between two rings which respects the operations of addition and multiplication.

More precisely, if R and S are rings, then a ring homomorphism is a function f : R → S such that

• f(a + b) = f(a) + f(b) for all a and b in R
• f(ab) = f(a) f(b) for all a and b in R
• f(1) = 1

Naturally, if one does not require rings to have a multiplicative identity then the last condition is dropped.

The composition

Function composition

In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

 of two ring homomorphisms is a ring homomorphism. It follows that the class

Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...

 of all rings forms a category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

 with ring homomorphisms as the morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

s (cf. the category of rings

Category of rings

In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings and whose morphisms are ring homomorphisms...

).

Properties

Directly from these definitions, one can deduce:

• f(0) = 0
• f(−a) = −f(a)
• If a has a multiplicative inverse in R, then f(a) has a multiplicative inverse in S and we have f(a⁻¹) = (f(a))⁻¹. Therefore, f induces a group homomorphism
  Group homomorphism
  In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
  
   from the (multiplicative) group of units of R to the (multiplicative) group of units of S.
• The kernel
  Kernel (algebra)
  In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
  
   of f, defined as ker(f) = {a in R : f(a) = 0} is an ideal in R. Every ideal in a commutative ring R arises from some ring homomorphism in this way. For rings with identity, the kernel of a ring homomorphism is a subring without identity.
• The homomorphism f is injective if and only if the ker(f) = {0}.
• The image
  Image (mathematics)
  In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
  
   of f, im(f), is a subring of S.
• If f is bijective, then its inverse f⁻¹ is also a ring homomorphism. f is called an isomorphism in this case, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings cannot be distinguished.
• If there exists a ring homomorphism f : R → S then the characteristic
  Characteristic (algebra)
  In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
  
   of S divides the characteristic of R. This can sometimes be used to show that between certain rings R and S, no ring homomorphisms R → S can exist.
• If R_p is the smallest subring
  Subring
  In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...
  
   contained in R and S_p is the smallest subring contained in S, then every ring homomorphism f : R → S induces a ring homomorphism f_p : R_p → S_p.
• If R is a field
  Field (mathematics)
  In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
  
  , then f is either injective or f is the zero function. Note that f can only be the zero function if S is a trivial ring or if we don't require that f preserves the multiplicative identity.
• If both R and S are fields
  Field (mathematics)
  In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
  
   (and f is not the zero function), then im(f) is a subfield of S, so this constitutes a field extension
  Field extension
  In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
  
  .
• If R and S are commutative and S has no zero divisors, then ker(f) is a prime ideal
  Prime ideal
  In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
  
   of R.
• If R and S are commutative, S is a field, and f is surjective, then ker(f) is a maximal ideal
  Maximal ideal
  In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
  
   of R.
• For every ring R, there is a unique ring homomorphism Z → R. This says that the ring of integers is an initial object
  Initial object
  In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
  
   in the category
  Category (mathematics)
  In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
  
   of rings.

Examples

• The function f : Z → Z_n, defined by f(a) = [a]_n = a mod n is a surjective ring homomorphism with kernel nZ (see modular arithmetic
  Modular arithmetic
  In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
  
  ).
• There is no ring homomorphism Z_n → Z for n > 1.
• If R[X] denotes the ring of all polynomial
  Polynomial
  In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
  
  s in the variable X with coefficients in the real number
  Real number
  In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
  
  s R, and C denotes the complex number
  Complex number
  A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
  
  s, then the function f : R[X] → C defined by f(p) = p(i) (substitute the imaginary unit i for the variable X in the polynomial p) is a surjective ring homomorphism. The kernel of f consists of all polynomials in R[X] which are divisible by X² + 1.
• If f : R → S is a ring homomorphism between the commutative rings
  Commutative ring
  In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
  
   R and S, then f induces a ring homomorphism between the matrix
  Matrix (mathematics)
  In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
  
   rings M_n(R) → M_n(S).
• The homomorphism f : Z → Z_n, defined by f(a) = [a]_n = a mod n followed by the inclusion mapping to Z is a ring homomorphism which is neither injective or surjective.

Types of ring homomorphisms

A bijective ring homomorphism is called a ring isomorphism. A ring homomorphism whose domain is the same as its range is called a ring endomorphism. A ring automorphism is a bijective endomorphism.

Injective ring homomorphisms are identical to monomorphism

Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....

s in the category of rings: If f:R→S is a monomorphism which is not injective, then it sends some r₁ and r₂ to the same element of S. Consider the two maps g₁ and g₂ from Z[x] to R which map x to r₁ and r₂, respectively; f o g₁ and f o g₂ are identical, but since f is a monomorphism this is impossible.

However, surjective ring homomorphisms are vastly different from epimorphism

Epimorphism

In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...

s in the category of rings. For example, the inclusion Z ⊆ Q is a ring epimorphism, but not a surjection. However, they are exactly the same as the strong epimorphisms.