Direct Sum of Rings: Explained

[BustedBreaks]

This may be a dumb question, but I just want to make sure I understand this correctly.
For [tex]R_{1}, R_{2}, ..., R_{n}[/tex]
[tex]R_{1} \oplus R_{2} \oplus, ..., R_{n}=(a_{1},a_{2},...,a_{n})|a_{i} \in R_{i}[/tex]

does this mean that a ring which is a direct sum of other rings is composed of specific elements of the original rings that satisfy distribution properties? That is, the first element of the new ring, a1, is from R1 etc for a2 to a_n. Is this correct?

 

[Landau]

Well, yes, you just wrote a_i \in R_i. As a set (forgetting the algebraic structure), the direct sum is just the direct product of sets, so the elements are ordered n-tuples with the i-th entry a_i from R_i, for all i=1,..,n.

 

[rayman123]

I have other question. What are the applications of the direct sum? Why is this a usefull constraction?