Prove that every ideal of a ring R is the kernal of a ring homorphism of R. In p

Question

Prove that every ideal of a ring R is the kernal of a ring homorphism of R. In particular, an ideal A is the kernel of the mapping r goes to r+A form R to R/A.

Explanation / Answer

Let A be an ideal in a ring R. Let f:R --> R/A is given by f(r)=r+A. The fact that f preserves addition and multiplication follows from the definition of addition and multiplication in R/A. It is surjective since any coset r+A is the image of r in R. Finally, the kernel is the set of all r in R such that f(r)=0+A, the zero element of R/A. But r+A=0+A r is congruent to 0 (mod A) r is in A. Thus, the kernel is just A.

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