Consider a commutative ring $A= ( \mathbb{R}^n , + , \times) $, where $+$ is the usual one. Assume further that $ \times $ is continuous (with respect to the usual topology). Let $H$ be the set of *non* invertible elements of this ring.

For $k \geq 0$, what is the largest integer $n=n(k)$ such that $\mathbb{R}^n$ can be endowed with a ring structure as described above, for which the corresponding $H$ is a vector space of dimension at most $k$ ?

For example, in the $k=0$ case we are looking for a field, and it is thus well-known that $n(0)=2$ (realized by $\mathbb{C} \simeq \mathbb{R^2}$). More generally, I can show that $n(k) \leq k + 2$ holds for any $k$ (hence showing that the quantity $n(k)$ is well defined !).

Is it true that $n(k) = k+2$ holds for all $k \geq 0$ ? In particular, is there a ring structure (as described above) on $\mathbb{R}^3$ such that $H$ is a line ?

Up to now, the only lower bound I have is the trivial $n(k) \geq n(k-1) \geq \cdots \geq n(0)=2$.