Consider the following condition on a ring:
For every nonzero $a$, there is a nonunit $b$ with the property that $a+b$ is a unit.
Observe that if $a$ is already a unit, then $b=0$ will do just fine. So the content of this condition is what it says about the nonzero nonunits of the ring.
Some rings satisfying the condition (I will write $\mathbf{1}$ for the multiplicative identity element of each).
For any $n \geq 1$ and field $F$, the polynomial ring $F[x_1, \dots, x_n]$.
For any $n \geq 1$ and field $F$, the ring $M_n(F)$ of $n \times n$ matrices over $F$.
Sketch of proof: Fix a nonzero $a \in M_n(F)$ and regard it as a linear transformation $T: F^n \to F^n$ in the obvious way. It is possible to find new bases $B$ and $B'$ of $F^n$ so that the matrix of $T$ with respect to these bases is a nonzero diagonal $\{0,1\}$ matrix $D$. (Extend a basis for $\ker T$ to a basis $B$ for $F^n$ by adding a list of vectors $B_0$; then note that $\{T(v): v \in B_0\}$ is a linearly independent subset of $F^n$, which you can extend to a basis $B'$ of $F^n$.) This implies the existence of invertible $U$ and $V$ in $M_n(F)$ with $a = UDV$. Take $b = U(\mathbf{1} - D)V$.
Any $C^*$-algebra $A$.
Proof: fix a nonzero $a \in A$. If the spectrum $\sigma(a)$ of $a$ contains a nonzero element $\lambda$ then we can take $b = \lambda \mathbf{1} - a$, so assume $\sigma(a) = \{0\}$. As $a$ is nonzero, one of $a + a^*$ and $a - a^*$ is nonzero; by replacing $a$ with $ia$ if necessary we may assume $a + a^*$ is nonzero. Since the spectral radius of a self-adjoint element coincides with its norm, there is a nonzero number $\lambda$ in $\sigma(a + a^*)$. Take $b = \lambda \mathbf{1} - (a + a^*)$. (The element $a + b = \lambda \mathbf{1} - a^*$ is invertible because $\lambda$ is nonzero and $\sigma(a^*) = \{\overline{z}: z \in \sigma(a)\} = \{0\}$.)
Some rings that do not satisfy the condition:
The ring of formal power series $F[[x_1, \dots, x_n]]$ (where $F$ is a field and $n \geq 1$).
More generally, any local ring.
For any $n > 1$ and field $F$, the ring of upper-triangular elements of $M_n(F)$.
Any commutative subalgebra of $\mathcal{B}(H)$ (the bounded operators on a Hilbert space $H$) containing a nonzero operator $a$ with $\sigma(a) = \{0\}$. (Due, e.g., to the nontrivial fact that if $b$ commutes with $a$ and $\sigma(a) = \{0\}$ then $\sigma(a + b) = \sigma(b)$.)
I don't know much ring theory. Is there a specific name for rings satisfying this condition?
If the answer to that is straightforward, here is a vaguer question: what does this condition "say" about a ring? Are there more familiar ring-theoretic properties it is related to? (For example: if you knew a lot of ring theory, and someone gave you a ring, how would you check whether or not it satisfied this condition?)
