Caleb M. answered • 02/01/17

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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!