Originally Posted by
FernandoRevilla That depends. For example, if you consider the "object" $\displaystyle 1$ as belonging to $\displaystyle V$ , it is a vector, and satisfies $\displaystyle 1++x=x++1=x$ for all $\displaystyle x\in V$ , so $\displaystyle 1$ is the zero vector. If you consider the "object" $\displaystyle 1$ as belonging $\displaystyle \mathbb{R}$ , it is a scalar and satisfies $\displaystyle a\cdot 1=1\cdot a=a$ , so $\displaystyle 1$ is the identity element for the product of scalars.
On the other hand, the "object" $\displaystyle 0$ does not belong to $\displaystyle V$ but belongs to $\displaystyle \mathbb{R}$ so, necessarily is an scalar, satisfying $\displaystyle a+0=0+a=0$ for all $\displaystyle a\in\mathbb{R}$ so, $\displaystyle 0$ is the identity element for the sum of scalars.