## Internal Direct Sum of Modules

**Point of Post: **In this post we discuss the notion of the internal directs sum of modules, giving the usual characterizations, etc. We also give a standard application to short exact sequences.

*Motivation*

We have seen that the category of modules has notions of coproducts, namely given a set of modules, we can form the module which is the submodule of the product of -tuples with finite support. That said, from experience dealing with vector spaces, we often see that coproducts sometimes come up more naturally as “internal direct sums”. Namely, a module may be able to be written as the sum of submodules in a “unique way” (i.e. unique representation). In this case, will be isomorphic to the coproduct over the set of modules, and so we don’t really get “new theory”. That said, often times we want to work with actual equalities: “ is the internal direct sum” opposed to “ is isomorphic to the coproduct”.

*Internal Direct Sum of Modules*

Let be a ring and a left -module. Suppose that is a family of submodules of . We clearly have, for each , the inclusion maps . We see then by definition of coproduct that we are afforded a guaranteed map . We say that is the *internal direct sum* of if is an isomorphism. Since, in an obvious sense, internal direct sums and coproducts are the same thing we shall use the same symbol to denote both. While it should be clear from context which we actually mean, the tipoff shall often come in the form of equality or isomorphism. For example, if one writes then its clear that we mean that is the internal direct sum of this family of modules (which are submodules).

Of course, while the above is a nice, succinct way of putting internal direct sums, it is often not how one actually thinks about the concept. No, really one often thinks about a module as being an internal direct sum of a family if the following characterization holds:

**Theorem: ***Let be a left -module and a family of submodules. Then, is the internal direct sum of if and only if every element of can be written uniquely as a sum with with all but finitely many .*

**Proof: **The fact that is equivalent to the statement that the canonical map is surjective, and the uniqueness of this representation is equivalent to the injectivity of .

For the case when we are dealing with only two submodules, this takes the following particularly nice form:

**Theorem: ***Let be a ring and a left -module. Then, if then if and only if and is trivial.*

**Proof: **Since we have a priori that it suffices to show that uniqueness of representation is equivalent the triviality of . Suppose first that we have uniqueness of representation. If , then we can write it as where we are thinking about and and where and . From uniqueness we may conclude that . Conversely, if is trivial and if then and so clearly and so and so uniqueness of representation follows.

In fact, it’s not hard to see that this generalizes to the following:

**Theorem: ***Let be a left -module and a family of submodules. Then, is the internal direct sum of if and only if and is trivial for each .*

*Relation to Short Exact Sequences*

We can use this notion of internal direct sums to be able to write the central module of a split short exact sequence in a nice way, in terms of the image and kernel of certain maps. First we note the following:

**Theorem: ***Let be a ring and left -modules. If and are -maps such that then .*

**Proof: **By the above theorems it suffices to show that is trivial and . To see the first, suppose that . Since we can find with . But, note then that says that but, since we see that and since we see that and so . Thus, . Since was arbitrary the triviality of follows. To see that we let be arbitrary. Note then that . Clearly and since we see that and so clearly . Since was arbitrary the conclusion follows.

Of course, applying this in conjunction with the splitting lemma gives us the following:

**Theorem: ***Let be a split short exact sequence. If is the guaranteed backmap for then .*

**References:**

[1] Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. *Advanced Modern Algebra*. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. *Module Theory.* Clarendon, 1990. Print.

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