Let R S be rings and fRS a homomorphism Suppose that S1 is a

Let R, S be rings, and f:RS a homomorphism. Suppose that S₁ is a subring of S. Prove that f^-1(S₁) = {r R : f(r) S₁} is a subring of R.

Solution

R and S are rings

S1 is a subring of S

As f from R to S is a homomorphsim f is both one to one and onto

Hence there exists inverse for S

S1 being a subring of S has inverse in R such that it is both one to one and onto.

As S1 is a subring, and f inverse is a homomorphism, it follows that R1 being the inverse of S1 satisfies

all the axioms of subring.

Because S1 is a subring S1 has multiplication identity e\'

If y is in S1 then f^-1(y) = x (say in R1)

then f^-1(y*e\') = f^-1(y)f^-1(e\')

= xe =x

Hence e is in R1 being inverse of e\' in S1