The question is fairly straight forward. I just found a question on the internet that asks whether all regular grammars areLL(1)LR(1)I guess they can't be LL(1) because of left recursion, but how do we prove that they are LR(1) if so.
Are regular grammars always LR(1)
context free;parsers
No.LR(k) grammars have to be unambiguous, that is there can be only one possible parse tree for every (valid) input. There are left- and right-regular grammars violating that, for example$\displaystyle\qquad S \to aS \mid aaS \mid a$or, if that one does not match your definition of right-regular, $\qquad\displaystyle\begin{align}  S &\to aS, S \to aA, \\  A &\to aA, A \to a\end{align}$as reinierpost mentions in a comment.Of course, every regular language does have an unambiguous (right-)regular grammar; get one from its minimal DFA using the standard construction.