If x has a left inverse w and a right inverse y they are equal. Write y = 1y = wxy = w(xy) = w1 = w. In this case x is called "invertible", or a "unit".
The set of units forms a group under multiplication. If R is the ring, the group of units is denoted R*. You've seen this before; Zn* is the group of units mod n.
Elements x and y are associates if x divides y and y divides x. show that "associate" forms an equivalence relation. Symmetry comes from the definition, and transitivity (x divides y and y divides z implies x divides z) is straightforward. Actually we need to be consistent here. Divides always means left divides, or right divides, as you prefer. Thus cx = y and ky = z, hence kcx = z.
We sometimes lump an element and all its associates together. In the integers, 5 and -5 are associates. When factoring 45, we don't spend a lot of time worrying about 5 verses -5. They are essentially the same prime factor; merely associates of each other. Of course 5 does not equal -5, and sometimes the particular associate does matter.
If xy = 0, and x and y are nonzero, x is a left zero divisor and y is a right zero divisor.
Suppose x has inverse w, and x is a zero divisor. Write 0 = xy = w(xy) = (wx)y = 1y = y, which contradicts x being a zero divisor. Invertible and zero divisor are mutually exclusive.
Assume every nonzero element is left invertible. Given v, write uv = 1, which makes u left and right invertible. Now u is a unit, with inverse v, and that makes v a unit with inverse u. The ring becomes a division ring.
A similar result holds for the right inverse, using similar algebra. If the right inverse of 1-yx is v, the right inverse of 1-xy is 1+xvy. Expand the product and replace yxv with v-1.
Next let R be the ring of endomorphisms on an infinite dimensional K vector space. Let b take the basis element ei to ei+1, and let a take ei to ei-1, with e1 mapping to 0. Now a*b (b followed by a) is the identity map, and a is a left inverse. However, a cannot be a right inverse, for running a first implies an endomorphism that is not injective.