I worry about “small” things.
Let us take a ring . It is easy to prove that that
. The standard proof works this way:
. Adding
on both sides, we get
.
This fact could be used to prove that . How?
. Hence, as
is the additive inverse of
, and as every element has a unique additive inverse, we have
.
Although proving the above is a fairly uninvolved exercise, it was never intuitive to me, unless I started to imagine as arithmetic addition and
as arithmetic multiplication. This blog post is an attempt to make the aforementioned facts more easy to visualize, and perhaps intuitive.
What is ? To answer this, we need to understand what
is in the context of distributivity of addition over multiplication. It is
. It is
. It is the combination of matter and anti-matter. It takes away as much as it gives. This I feel is the easiest way to think about it in the given context. I would love to elaborate on why this is the best way some other time.
Now what is ? It is
. It is
. If
, then
. This proves that
, as the additive inverse of every element is unique.
Using this fact, we can now say that
.
All this because I find it easier to visualize as
for any element
rather than
. And I’m sure you’d find it easier too.