Definition of Direct Sum of Ideals

Question

I've been searching the internet, and I can't find a definition for the direct sum of ideals. In a previous question I posted, the author writes $M_n(D) =\oplus I_R$, where the $I_R$ are subrings and left ideals of the matrix ring $M_n(D)$, and are formed by the matrices $M=[m_{ij}]$ with $m_{ij}=0$ for $j\neq r$. $D$ is a division algebra. Is the direct sum only defined for rings, so that we can only consider the direct sum because they are subrings? Any clarification would be well appreciated!

Answer 1

The (internal) direct sum is not "only defined for rings." There are analogous versions for other algebraic objects including groups, modules and others.

It turns out that the underlying set of the direct sum of ideals is the same no matter which of the following ways you think of it:

• a direct sum of subgroups of an abelian group
• a direct sum of (non unital) subrings of $R$
• a direct sum of submodules $R_R$
• a direct sum of sub-bimodules of $_RR_R$

The first one is the most general and is already enough: $\oplus_{i\in I}I_i$ is the set of all finite sums of the form $\sum x_i$ with $x_i\in I_i$. It only makes reference to addition, and no module structure.

However, this direct sum already does have a module (in fact a bimodule) structure, since each term in the sum absorbs left and right multiplication into its respective ideal. Ideals are by definition sub-bimodules of $_RR_R$, so in some sense it is natural to think of them as bimodules summing to another sub-bimodule of $R$.