For a direct sum of modules, can the two modules be the same? I was referring to this post where the author claims that if $R=\oplus_i N_i$ a direct sum of simple left ideals, some of the $N_i$ may repeat.
Is that possible? I mean, by definition of direct sum shouldn't each $N_i$ be necessarily distinct as they have trivial intersection?
Thanks.
 A: A good example is $\mathbb{Z}\oplus \mathbb{Z}$.  Each component is isomorphic, but $(0,3)$ is different than $(3,0)$.  That is what is meant by each summand being unique.
A: By $N_1 \oplus N_2 \oplus ... N_k$ we actually mean $\bar{N_1} \oplus \bar{N_2} \oplus ... \bar{N_k},$
where  $$\bar{N_1} = (x,0,0,...,0)|x \in N_1 \},$$ $$.$$ $$.$$
$$\bar{N_i} = (0,0,...,x,...,0)|x \in N_i \}.$$ We can see that  $\bar{N_i} \cong N_i$ and for distinct $i, j$ we always have $\bar{N_i} \cap \bar{N_j} = 0$, even if $N_i = N_j$.
(Edit: By $N_1 \oplus N_2 \oplus ... N_k$ some authors denote $N_1 + N_2 + ... N_k$ but only when the sets happen to be disjoint.)
A: In this context, "the same" means "isomorphic as $R$-modules", not "equal as subsets of $R$".  Of course, they cannot be equal as subsets of $R$ for the reason you point out.
On a related point, the term "direct sum" actually has two different meanings.  For simplicity, I will explain them only for binary direct sums.  The first meaning is an "internal direct sum": a module $M$ is the internal direct sum of two submodules $N_1,N_2\subseteq M$ iff $N_1\cap N_2=0$ and $N_1+N_2=M$.  The second meaning is an "external direct sum": if $N_1$ and $N_2$ are two modules, their external direct sum is the module $N_1\oplus N_2$ (or more generally, any isomorphic module) whose underlying set is cartesian product $N_1\times N_2$ and whose module structure is defined separately on each coordinate.  The connection between the two definitions is that whenever $N_1$ and $N_2$ are submodules of $M$, there is a canonical homomorphism from the external direct sum $N_1\oplus N_2$ to $M$, and this homomorphism is an isomorphism iff $M$ is the internal direct sum of $N_1$ and $N_2$.  And conversely, the external direct sum $N_1\oplus N_2$ is the internal direct sum of its two submodules $N_1'=N_1\times \{0\}$ and $N_2'=\{0\}\times N_2$ which are isomorphic to $N_1$ and $N_2$.
In an internal direct sum, obviously $N_1$ and $N_2$ must be distinct unless they are both $0$ (since $N_1\cap N_2=0$).  However, when defining the external direct sum, it is perfectly fine if $N_1$ and $N_2$ are equal, as discussed in the other answers.  
A: In general, a direct sum of modules $M$ and $N$ is really the Cartesian product $M\times N$ with the added natural structure of a module. With this structure, you denote the Cartesian product by $M\oplus N$, and as Joe Johnson 126 pointed out, there is no need for $M$ and $N$ to be distinct.
