So I'm learning about Ring Theory and have gotten to Ideals. My book tells me that the product of two ideals I and J or a ring R (standard * and + operations) is the set of all finite sums of elements of the form ij where i is in I and j is in J. I'm having trouble coming up with an example as to why we can't just define the product of I and J to be just the elements of the form ij (not any finite sum).(adsbygoogle = window.adsbygoogle || []).push({});

The definition without the sum seems to work for the rings i've tried (R being the integers Z, and ideals being nZ and mZ for different m and n, for example). I suspect it breaks down with matrices, or maybe polynomials, but i'm having trouble nailing down a specific example. Guidance would be much appreciated. Thanks!

- JD

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# Product Of Ideals - Why is the sum necessary?

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