I have some questions about definitions from Otto Forster's Lectures on Riemann Surfaces.
The setup: Let $X$ be a Riemann surface, and allow $U \subset X$ to be open. Define $\mathcal{E}(U)$ to be collection of all functions $f:U\to\mathbb{C}$ such that $f$ is differentiable in the real sense, i.e. by realizing $z=a+ib$ as an the element $(a,b) \in \mathbb{R}^2$. Define $\mathfrak{m}_a = \{ f\in\mathcal{E}(U): f(a) = 0\}$, and $\mathfrak{m}^{2}_{a} = \{f \in \mathfrak{m}_a : f_x(a) = f_y(a) = 0\}$.
Question one: what is meant by saying $\mathcal{E}(U)$ is a $\mathbb{C}$-algebra? Does it simply mean that we can add and distribute multiplication of functions pointwise with the underlying rules determined by $\mathbb{C}$?
Question two: what are the elements of the cotangent space $T^{(1)}_{a} = \frac{\mathfrak{m_a}}{\mathfrak{m}^{2}_{a}}$? Meaning, what are the equivalence classes of this quotient space? Is there an explicit example of a cotangent space that is easy to grasp? I'm unfamiliar with quotients by (sub-)algebras. I'm really having difficulties visualizing or understanding this definition which is vital for the rest of the chapter.