divisible modules In surveying LMR of T.Y.Lam, I reached a paragraph stating that "when R is a domain every direct sum or direct product of divisible modules is divisible." My question is that "should R is not a domain, might it be possible to have a divisible direct sum ( or product ) of divisible modules?" 
Not being able to find any (dis-)proof, I so appreciate any one answering my question.

You will have to read the text more closely for the definition of "divisible" that Lam uses. He uses a definition that is not very ordinary, but he makes the case for it very well.
If I'm recalling correctly, given a right $R$ module $M$, we say $M$ is divisible if for every $m\in M$ and $x\in R$ with $r.ann(x)\subseteq ann(m)$, there exists $n\in M$ such that $nx=m$.
The annihilator condition is obviously necessary for this to occur, and of course in a domain $r.ann(x)=0$ for every nonzero $x$, and the definition becomes the usual definition of divisibility for domains.
I can't remember all of the justifications given offhand, but I do remember two nice things that work out with this definition: every injective module is divisible, and a ring whose modules are all divisible is von Neumann regular.
I seem to remember it is somehow dual to another natural condition, but I cannot rightly remember what the statement was at the moment.
