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## Main Question or Discussion Point

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).

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

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