Pumping Lemma on Language with subtracted length My study group and I have had some back and forth on one exercise and I haven't found any matching solution online. The task looks as follows: Prove that $L$ is not regular given

$$ L = \{ a^k b a^{m-1} \mid k,m \in \mathbb{N} \} $$

Questions:


How does $m-1$ affect my choice of the word to be pumped? There must be a way to rule out $m=0$ because otherwise my length would be negative. 
Would it be wrong to choose $w = a^{n-1}ba^n$ with the intent of pumping down $y = b$?  


Every answer is highly appreciated.
You could argue with a reasonably straight face that L is not well-defined, since it supposedly contains the word $a^0ba^{-1}$, so L wouldn't be a language at all, therefore not a regular language and not an irregular language either. 

If you change it slightly to $$ L = \{ w: w = a^k b a^{m-1} \mid k,m \in \mathbb{N} \} $$ then it is definitely regular; there's a very simple FSM with just two states for it.