Let $R$ be an unital associative ring and let $f: F \rightarrow R$ be an onto ring homomorphism. Where F is some freely generated ring over the set $S$ then $<S|T>$ is called presentation of ring $R$ where $T$ is minimal set of relation.
What does it mean by relation $T$? also is surjectivity essential there?
Any help/hint in this regards would be highly appreciated. Thanks in advance!
You stated in the comments that you knew about group presentations. Well here it's the same thing but with rings :
If you say that $\langle S\mid T\rangle$ is a presentation for a group $G$ it means that $G$ is the quotient of the free group generated by $S$ by the normal subgroup generated by $T$ ($T$ is a set of words on $S$).
If now you say that $\langle S\mid T\rangle$ is a presentation of the ring $R$, it means that $R$ is the quotient of the free ring generated by $S$ by the (bilateral) ideal generated by $T$. In other words, elements of $T$ are elements of the free ring on $S$, and in $\langle S\mid T\rangle$ you declare that these are $0$.
For instance $\langle x,y\mid xy-yx\rangle$ will be (isomorphic to) the usual commutative polynomial ring $\mathbb{Z}[x,y]$.
Surjectivuty is essential because otherwise $R$ is not generated by $S$ !