Question: Given two matrices $A\_{m\times n}$ and $B\_{n\times p}$, what is the sufficient and necessary condition for $AB$ to have full rank?

I know $r(AB)=r(B)-\dim N(A) \cap R(B)$, so is it true the above iff $\dim N(A)\cap R(B)=0$? But seems incorrect if $m\lt p$. Please help.

Answer: 
The indefinite integral $\int f(x) dx$ denotes, in this particular context, the *set* of all primitives of $f(x)$. If you find one primitive, say $F(x)$, then you know all other primitives have the form $F(x) + C$, where $C$ is any constant. This is what we mean when we write $\int f(x) dx = F(x) + C$: the set of all primitives of $f(x)$ is the set of all the functions of the form $F(x) + C$.
The "integration constant" $C$ does have the "purpose" to make a seemingly true equation at least halfway true. In fact it is what many people call a "dangling variable", similar to the $i$ and $k$ when we talk about a "matrix $\bigl[a\_{ik}\bigr]\ $". As noted in other answers, an *indefinite integral* is not a function, but an infinite set of functions, any two of them differing by a constant. Writing $\langle F(x)\rangle$ (or similar) instead of $F(x)+C$ for the set of all functions differing from the term $F(x)$ by a constant, one could write, e.g.,
$$\int (5x -6x^4)\ dx =\langle {5\over2} x^2-{6\over5} x^5\rangle\ ,$$
and the mysterious constant has disappeared.