1. ## Ring

Let R be a ring, and I be an ideal of G.
Show that there is a one to one and onto correspondence between the subrings of R/I and those subrings of R which contain I.
Hint: maybe the "natural" homomorphism f: R-> R/I defined by f(x)= x+I

2. Originally Posted by evonnova
Let R be a ring, and I be an ideal of G.
Show that there is a one to one and onto correspondence between the subrings of R/I and those subrings of R which contain I.
Hint: maybe the "natural" homomorphism f: R-> R/I defined by f(x)= x+I

Define $\displaystyle \pi: R\mapsto R/I$ to be the natural projection. If $\displaystyle S$ is a subring of $\displaystyle R$ then $\displaystyle \pi (S)$ is a subring of $\displaystyle R/I$. If $\displaystyle K$ is a subring of $\displaystyle R/I$ then $\displaystyle \pi^{-1} (K)$ is a subring of $\displaystyle R$ which contains $\displaystyle R$.
Let $\displaystyle X$ be the set of all subrings of $\displaystyle R$ which contain $\displaystyle I$ and $\displaystyle Y$ be the set of all subrings of $\displaystyle R/I$. By above paragraph define $\displaystyle \hat \pi: X\mapsto Y$ as $\displaystyle \hat \pi (x) = \pi (x)$ for all $\displaystyle x\in X$.