My abstract book defines inverses (units) as solutions to the equation $ax=1$ then stipulates in the definition that $xa=1=ax$, even in non-commutative rings. But I'm having trouble understanding why this would be true in the generic case.
Can someone help me understand?
Book is "Abstract Algebra : An Introduction - Third Edition" By Thomas W. Hungerford. ISBN-13: 978-1-111-56962-4 (Chapter 3.2)
Edit:
Ok, I think I got it. Left inverses are not necessarily also right inverses. However, if an element has a left inverse and a right inverse, then those inverses are equal:
$$ lx = 1 \\ xr = 1\\ lxr = r \\ lxr = l \\ r = l $$
Source:http://www.reddit.com/r/math/comments/1pdeyf/why_do_multiplicative_inverses_commute_even_in/cd17pcb
Also, I assumed that 'unit' was synonymous with 'has a multiplicative inverse'. It is not; a 'unit' is an element that has both a left and right inverse, not just an inverse in general.
No, I have the book in front of me and he defines units as elements that have both a left and a right inverse, so $a$ is a unit if there exist elements $x$ and $y$ in $R$ such that $ax=ya=1$. Note $x$ need not equal $y$.
In his very next Remark he then proves that in this situation $x=y$, as a theorem, not as part of the definition.
Edit: Now you are editing your question and filling it with even more confusion. You might try a book with more examples, such as Dummit & Foote.