principal ideal ring

Question

Suppose that

R=R1⊕R2⊕…….⊕Rn
where each Ri is a principal ideal ring. Verify that R is also a principal ideal ring.

 

please I want help with that

 

Answer 1

In a PID, every ideal is generated by a single element. It suffices to consider the case when n=2 (the general case follows quickly mutatis mutandis). If R= R_1 \oplus R_2, and I is an ideal of R, then (1,0)I and (0,1)I are ideals of R_1,R_2. What can you say about (1,0)I and (0,1)I then? Does this give you an element generating R?

 

Note that this tells you what the ideals of a direct sum look like generally - so long as the ring has unity and the sum is direct!