Ring Homomorphism - Examples

Examples

• The function f : Z → Z_n, defined by f(a) = _n = a mod n is a surjective ring homomorphism with kernel nZ (see modular arithmetic).
• There is no ring homomorphism Z_n → 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 M_n(R) → M_n(S).