Abstract Algebra: Theory and Applications · Problem 39

Polynomials — Thomas Judson

Mathematics — Abstract Algebra · chinese-remainder-theorem-rings

Problem

The Chinese Remainder Theorem for Rings. Let

R

be a ring and

I

and

J

be ideals in

R

such that

I + J = R

. (a) Show that for any

r

and

s

R

, the system of equations

\begin{aligned} x &\equiv r \pmod{I}\\ x &\equiv s \pmod{J} \end{aligned}

has a solution. (b) In addition, prove that any two solutions of the system are congruent modulo

I \cap J

. (c) Let

I

and

J

be ideals in a ring

R

such that

I + J = R

. Show that there exists a ring isomorphism

R/(I \cap J) \cong R/I \times R/J.