Question

Which of the following statements are true? Justify your answers. (This means that if you

think a statement is false, give a short proof or an example that shows it is false. If it is

true, give a short proof for saying so.)

(i)Given any ring R, there is an ideal I of R such that R/I is commutative.

(ii)If S is an ideal of a ring R and f a ring homomorphism from R to a ring R', then f-¹(f(S))=S.

Answer 1

(i) It is true that for any ring $R$, there is an ideal $I$ of $R$ such that $R/I$ is commutative. Indeed, let $I=R,$ then the quotient ring $R/I=R/R$ is singleton, and hence is commutative.

(ii) It is false that if $S$ is an ideal of a non-trivial ring $R$ and $f$ a ring homomorphism from $R$ to a ring $R',$ then $f^{-1}(f(S))=S.$ Indeed, let us consider the trivial ring homomorphism $f:R\to R',\ f(x)=0'$ for each $x\in R.$ Let $S=\{0\}$ be a trivial ideal of $S.$ Then $f^{-1}(f(S))=f^{-1}(f(\{0\}))=f^{-1}(0')=R\ne\{0\}=S.$

It is true only for trivial ring $R=\{0\}.$ In this case a unique ideal of $R$ is $S=\{0\}=R$ and for any homomorhism $f$ we have that $f^{-1}(f(S))=f^{-1}(f(\{0\}))=f^{-1}(0')=\{0\}=S.$