Do random variables follow the same algebraic rules as ordinary numbers? In the comments on my answer to a recent question about the sum of random variables, I came across a link to the Wikipedia article on the ratio distribution, and noticed the following peculiar claim there:

The algebraic rules known with ordinary numbers do not apply for the algebra of random variables. For example, if a product is $C = AB$ and a ratio is $D=C/A$ it does not necessarily mean that the distributions of $D$ and $B$ are the same.

This claim has been in the article since 2007.  It was added there by the same seemingly reputable editor who originally created the article and contributed much of its original and current content, and it is seemingly cited to the book The Algebra of Random Variables by Melvin D. Springer, published in 1979 (although it's not 100% clear whether or not the citation marker that appears later in the same paragraph is actually meant to cover this claim as well).

Obviously, the claim seems like nonsense to me.  I could just edit it out of the Wikipedia article, but given that it has stood unchallenged there for over 10 years, I figured I should make sure I'm not the one who's mistaken here.
Not having Springer's book at hand to check the (possible) citation, I figured I'd ask the experts here for help.  In particular, since the claim as stated really consists of two parts, so does my question:
Part 1: Do random variables follow the same algebraic rules as ordinary numbers, or do they (in some sense) not?  If they do not, how do the rules differ?  Does it depend on what (generally accepted) formalism one adopts?
Part 2: It is clear that, even for ordinary numbers, $D = \frac{AB}{A}$ is not always equal to $B$, since $D$ is not even defined when $A = 0$.  Is this trivial difference the only way in which $D$ and $B$ can fail to be equal, even when they are random variables?  In particular, does the following statement always hold for (real- or complex-valued) random variables: $$A \ne 0 \implies \frac{AB}{A} = B.$$
Part 3 (bonus): What does Springer's book actually say about this, and is there anything in there that could in some sense be taken to support the claim quoted above?  Is it, as I would presume, actually regarded as a reliable source for claims about mainstream mathematics and statistics?
 A: The algebra of random variables (ARV) is an extension of the usual algebra of numbers "high school algebra". This must be so because numbers can be embedded in the ARV as rv equal to a constant with probability 1. So there cannot be any inconsistency, but it could well be new properties which doesn't say anything about numbers. In the ARV equality is equality in distribution, so it is really an algebra of distributions. But for rv's constant with probability 1, this is an extension of equality of numbers in the usual sense. 
About the given example from Wikipedia, there is no inconsistency there, only a (maybe for someone) surprising possibility that arises because there are many random variables such that $X$ and $X^{-1}$ have the same distribution, while there are only two numbers with this property, $-1$ and 1. The Cauchy distribution have this property, see What can we say about distributions of random variables $X$ such that $X$ and its inverse $1/X$ have the same distribution?.  
A: Random variables are actually functions (measurable on a sample space) so they follow the rules of functions.  Confusion comes in what "=" means since it is often misused as "same distribution" in practice rather than truly identical.
