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Homomorphism and Subrings 
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#1
Oct2810, 05:56 PM

P: 3

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(xy) = 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? 


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