Iwahori-Hecke algebra of the symmetric group

Definition
Let $$R$$ be a unital ring and $$n$$ be a natural number. The Iwahori-Hecke algebra of the symmetric group of degree $$n$$ over the ring $$R$$ is defined as the $$R[q]$$-algebra given by the presentation:

$$\langle T_1, T_2, \dots T_{n-1} \mid (T_i - q)(T_i + 1) = 0, T_iT_j = T_jT_i \ \forall |i - j| > 1, T_iT_{i+1}T_i = T_{i+1}T_iT_{i+1} \rangle$$.

In other words, it is the Iwahori-Hecke algebra corresponding to the symmetric group of degree $$n$$, viewed as a Coxeter group, in the usual way: Symmetric group on a finite set is a Coxeter group.

For a field with $$q$$ elements, specializing to the value $$q$$ gives the Hecke algebra of the general linear group $$GL_n(\mathbb{F}_q)$$.