Ring Homomorphism - Examples

Examples

  • The function f : Z → Zn, defined by f(a) = n = a mod n is a surjective ring homomorphism with kernel nZ (see modular arithmetic).
  • There is no ring homomorphism Zn → Z for n > 1.
  • If R 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 → 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 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 Mn(R) → Mn(S).

Read more about this topic:  Ring Homomorphism

Famous quotes containing the word examples:

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)

    “It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
    —G.C. (Georg Christoph)

    “There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
    —Bernard Mandeville (1670–1733)