In a field, if we require that all polynomials have at least one root, then it's algebraically closed and all polynomials factor completely. In a ring, the same requirement implies that it's an ACF, because linear polynomials need a root and this ensures multiplicative inverses. This raises the question, is there some other way a ring can be "almost" algebraically closed, without turning completely into a field?
So my question: Is there a ring, in which all polynomials of degree $\ge 2$ have a root, that is not also a field?
Naturally we can't require that all polynomials of degree $\ge 2$ have as many roots as their degree, otherwise one of the two roots of $$ax^2 - (a-1)x + 1 = (x-1)(ax-1)$$ would give us an inverse of $a$.