TITLE: Is the matrix ring $\mathrm{Mat}_n(\mathbb{C})$ "algebraically closed"?
QUESTION [8 upvotes]: In spite of the fact that the matrix ring $\mathbb{C}^{n \times n}$ is not a field, is it still possible to talk about it being 'algebraically closed' in the sense that $\forall f \in \mathbb{C}^{n \times n}[x]$ does $\exists A \in \mathbb{C}^{n \times n}$ such that $f(A) = 0$? If so, then is it 'algebraically closed'?
Are there any other non-field sets that this idea can be extended to?

REPLY [11 votes]: I'd like to add that a nice theory of roots of polynomials over noncommutative rings was developed by I. Gelfand, V. Retakh, and R. Wilson, see the paper arXiv:math/0208146 and references therein (in particular, the earlier paper by Gelfand and Retakh on the noncommutatoive Vieta theorem).