In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

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 of two ring homomorphisms is a ring homomorphism. In Mathematics, a composite function represents the application of one function to the results of another It follows that the class of all rings forms a category with ring homomorphisms as the morphisms (cf. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and the category of rings). In Mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity and whose Morphisms

Contents 1 Properties 2 Examples 3 Types of ring homomorphisms 4 See also

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 from the group of units of R to the group of units of S. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function
• The kernel of f, defined as ker(f) = {a in R : f(a) = 0} is an ideal in R. 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 In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. 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 of f, im(f), is a subring of S. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage
• If f is bijective, then its inverse f⁻¹ is also a ring homomorphism. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 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 of S divides the characteristic of R. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without 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 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. In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations
• If R is a field, then f is either injective or f is the zero function. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division (Note, however, that if f preserves the multiplicative identity, then it cannot be the zero function. )
• If both R and S are fields, then im(f) is a subfield of S (if f is not the zero function). In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division
• If R and S are commutative and S has no zero divisors, then ker(f) is a prime ideal of R. In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers
• If R and S are commutative, S is a field, and f is surjective, then ker(f) is a maximal ideal of R. In Mathematics, more specifically in Ring theory, a maximal ideal is an ideal which is maximal (with respect to set inclusion amongst all proper ideals
• For every ring R, there is a unique ring homomorphism Z → R. This says that the ring of integers is an initial object in the category of rings. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships

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). In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers
• There is no ring homomorphism Z_n → Z for n > 1.
• If R[X] denotes the ring of all polynomials in the variable X with coefficients in the real numbers R, and C denotes the complex numbers, 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. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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 R and S, then f induces a ring homomorphism between the matrix rings M_n(R) → M_n(S). In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

Types of ring homomorphisms

• A bijective ring homomorphism is called a ring isomorphism. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property
• A ring homomorphism whose domain is the same as its range is called a ring endomorphism.

Injective ring homomorphisms are identical to monomorphisms 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. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. 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 epimorphisms in the category of rings. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which For example, the inclusion Z ⊆ Q is a ring epimorphism, but not a surjection. However, they are exactly the same as the strong epimorphisms.

See also

• homomorphism

In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector

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