Homomorphism and Subrings

  1. 1. The problem statement, all variables and given/known data

    Let f: R -> S be a homomorphism of Rings and T a subring of S.
    Let P = { r belongs to R | f(r) belongs to T}
    Prove P is a subring of R.

    2. Relevant equations
    Theorems used:
    If S and nonempty subset of R such that S is closed under multiplication and addition, then S is a subring of R.

    If f : R -> S a homomorphism of rings, then f(OR) = 0S
    (0R is the 0th element of R and similar for 0S)

    3. The attempt at a solution

    First i showed P nonempty. R is a ring So O(R) belongs to R. Then f(0R) = 0S because f is a homomorphism and f maps the zero element to the zero element ( prevevious result)

    But T is a Subring of S so 0S belongs to T thus P is nonempty.

    (There is a theorem that says if I show P is nonempty I just now show closure under subtraction and multiplication to show P is a subring)

    So let x and y belong to P
    Now f(x-y) = f(x) - f(y). Doesn't this have to belong to T?
    Both f(x) and f(y) are in T since each x and y belong to P
    But because T is a subring of S isn't it closed under subtraction already so f(x) - f(y) belongs to T?

    Then f(xy) = f(x)f(y) and a similar argument holds?
  2. jcsd
  3. micromass

    micromass 18,533
    Staff Emeritus
    Science Advisor

    This is entirely correct. And the multiplication is indeed analogous.
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