Detailed solutions please 1 Let R and S be commutative rings

Detailed solutions please: 1. Let R and S be commutative rings and let sigma: R right arrow S be a homomorphism. Prove that if sigma is one to one and b in R is a zero divisor of R, then sigma(b) is a zero divisor of S.

Solution

we are given that R--->S is a homomorphism

A homomorphism is a map that preserves selected structure between two algebraic structures, with the structure to be preserved being given by the naming of the homomorphism

the set R and S are the same as they form a homomorphic relation.

now b E R that is b lies with the set R

and since R and S are homomorphic so b will also lie in the ser S

since sigma is one to one function threfore the entire domain of sigma will be mapped .

its also gven that sigma(b) is a zero divisor of R

and since b E S as well

so sigma(b) is a zero divisor of S as well

hence proved.