Let $k⟨X⟩$ be a free associative algebra generated by a set $X$ over a field $k$. Let $S$ be a set of $k$-algebra relations. Then what does it mean by the set of relations are independent ? 



Independent set of relations in an algebra
Do you look $k<X>$ as a right module (over $k<X>$ itself) ? or a left-right bimodule ? (which seems to indicate $k$-algebra relations, look at the comment of Todd Trimble). 


There is a theorem that I used many times to construct minimal automata (with multiplicities) and reduce the set of relations in a presented algebras, it is the following (th 3.2 p35 in Berstel and Reutenauer's book *Rational series and their languages* ,Monographs in Theoretical Computer Science. An EATCS Series, 1988). I can elaborate if you give me more contextual precisions (is the quotient finite-dimensional ?) 


**Theorem** Let $I$ be a right ideal of $k<X>$ . There exists a prefix code
$C$ with associated prefix-closed set $P$, and coefficients 
$$
\alpha(c,p)\in k;\ (c \in C, p \in P )
$$
 such that the polynomials 
 $$
 (P\_c = c − \sum\_{p\in P} \alpha(c,p) p) (c ∈ C) 
 $$
 generate freely $I$ as a right $k<X>$ -module and such that P defines a $k$-basis 
 in $k<X>/I$.
Is it what you are looking for ? (if this is the case, I can give algorithms)


