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Let $F$ be a field and $(K,\bullet)$ be a unital $F$-algebra. For each $v \in K$ let $S(v)$ be the set of all $a \in F$ satisfying the condition that $v -a.1_K $ does not have an inverse with respect to the operation $\bullet$. If $v \in K$ has a multiplicative inverse $v^{-1}$ with respect to this operation, show that either $S(v) = \emptyset = S(v^{-1})$ or $S(v) \neq \emptyset$ and $S(v^{-1}) = \{ a^{-1} \text{ where }a \in S(v) \}$.

If $K$ is associative then the problem becomes easy, and in that case $v^{-1}$ is uniquely defined. However I am stuck on the case $K$ isn't associative.

The problem is from Golan's book on Linear Algebra (Third Ed, Exercise 131), and an inverse of $v \in K$ is defined as an element $u \in K$ satisfying $u \bullet v = v \bullet u = 1_K$.

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