## Direct Sum of Algebra

**Point of post: **In this post we shall discuss a natural way to build new algebras out of a collection of algebras. This is the direct sum of algebras which, unsurprisingly, mimics the construction of the direct sum of groups or direct sum of vector spaces.

*Motivation*

Just as is the case for vector spaces, groups, modules etc. one can define the direct sum of algebras.

**Direct Sum of Algebras**

Namely, if is a collection of -algebras and we define the *support *of to be the set

We then define the *direct sum of , *denoted , to be the set of all elements of with finite support with coordinate wise operations. More explicitly if we denote in by respectively (where we evidently mean and ) then we define addition in by , multiplication by , and scalar multiplication for . In other words, we just give the ring structure of the direct sum of rings and just multiply scalars coordinate wise. It’s evident that this does, in fact, define an -algebra on with unity equal to .

It’s clear that if is in fact finite that this reduces to defining coordinate wise operations on the set .

For now it will suffice to know just the definition of the direct sum of algebras, but just because they are easy we prove a fundamental results. Namely,

**Theorem: ***Let be a field and a collection of -algebras then *

* *

**Proof: **Let be an element of then evidently for any in we have that so that is a member of . Conversely, if is an element of then we evidently have for each and that where for and is an element of and thus so that for every and in particular . Since was arbitrary it follows that is an element of and thus since was arbitrary we have that is an element of , but since all but since is finite we must have that it, in fact, an element of . The conclusion follows.

**References:**

Roman, Steven. *Advanced Linear Algebra*. New York: Springer, 2005. Print.

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