Prove the Correspondence Theorem: Let I be an ideal of a ring R. Then S ? S/I is a one-to-one correspondence between the set of subrings S containing I and the set of subrings of R/I. Furthermore, the ideals of R correspond to ideals of R/I.&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;Let p be prime. Prove thatZ(p) ={a/b: a,b?Zandgcd(b,p)=1}

Question

Prove the Correspondence Theorem: Let I be an ideal of a ring R. Then S ? S/I is a one-to-one correspondence between the set of subrings S containing I and the set of subrings of R/I. Furthermore, the ideals of R correspond to ideals of R/I.&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;&amp;gt;Let p be prime. Prove thatZ(p) ={a/b: a,b?Zandgcd(b,p)=1}

Maggie Tanner · Accepted Answer

clearly S? T is a subset of S and T both and so asunset of R.To show that it is a subring of R we have to show thatfor any u, v ? S? T(1) u - v ? S? T(2) u.v ? S? Tproof (1)Since u, v ? S? T =&amp;gt; u, v ? Sand u, v ? TNow S and T are subring of R =&amp;gt; u - v ? S and u- v ? T=&amp;gt;u - v ? S? Tproof (2)Since u, v ? S? T=&amp;gt; u, v ? S and u, v ? TNow S and T are subring of R =&amp;gt; u.v ? S and u.v? T=&amp;gt;u.v ? S? THence S? T is a subring of R

Answered question

Answer & Explanation

New Questions in Pre-Algebra