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.