Ring homomorphism

From Academic Kids

In abstract algebra, a ring homomorphism is a function 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 : RS 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

(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. It follows that the class of all rings forms a category with ring homomorphisms as the morphisms.

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.
  • The kernel of f, defined as ker(f) = {aR : f(a) = 0} is an ideal in R. Every ideal in R arises from some ring homomorphism in this way. f is injective if and only if the ker(f) = {0}.
  • 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 Rp is the smallest subring contained in R and Sp is the smallest subring contained in S, then every ring homomorphism f : RS induces a ring homomorphism fp : RpSp. This can sometimes be used to show that between certain rings R and S, no ring homomorphisms RS can exist.
  • For every ring R, there is a unique ring homomorphism ZR. This says that the ring of integers is an initial object in the category of rings.

Examples

  • The function f : ZZn, defined by f(a) = [a]n = a mod n is a surjective ring homomorphism with kernel nZ (see modular arithmetic).
  • There is no ring homomorphism ZnZ 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. The kernel of f consists of all polynomials in R[X] which are divisible by X2 + 1.
  • If f : RS is a ring homomorphism between the commutative rings R and S, then f induces a ring homomorphism between the matrix rings Mn(R) → Mn(S).

See also

Navigation

Academic Kids Menu

  • Art and Cultures
    • Art (http://www.academickids.com/encyclopedia/index.php/Art)
    • Architecture (http://www.academickids.com/encyclopedia/index.php/Architecture)
    • Cultures (http://www.academickids.com/encyclopedia/index.php/Cultures)
    • Music (http://www.academickids.com/encyclopedia/index.php/Music)
    • Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
  • Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
  • Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
  • Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
    • Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
    • Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
    • Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
    • Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
  • History (http://www.academickids.com/encyclopedia/index.php/History)
    • Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
    • Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
    • Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
    • Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
    • Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
    • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
    • United States (http://www.academickids.com/encyclopedia/index.php/United_States)
    • Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
    • World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
  • Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
  • Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
  • Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
  • Science (http://www.academickids.com/encyclopedia/index.php/Science)
    • Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
    • Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
    • Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
    • Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
    • Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
    • Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
    • Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
    • Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
  • Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
    • Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
    • Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
    • Government (http://www.academickids.com/encyclopedia/index.php/Government)
    • Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
    • Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
  • Space and Astronomy
    • Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
    • Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
  • Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
  • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
  • Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
  • US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

  • Home Page (http://academickids.com/encyclopedia/index.php)
  • Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

  • Clip Art (http://classroomclipart.com)
Toolbox
Personal tools