Can a regular grammar be ambiguous? An ambiguous grammar is a context-free grammar for which there exists a string that has more than one leftmost derivation, while an unambiguous grammar is a context-free grammar for which every valid string has a unique leftmost derivation.  

A regular grammar is a mathematical object, $G$, with four components, $G = (N, \Sigma, P, S)$, where 


$N$ is a nonempty, finite set of nonterminal symbols, 
$\Sigma$ is a finite set of terminal symbols, or alphabet, symbols, 
$P$ is a set of grammar rules, each of one having one of the forms:  


$A \rightarrow aB$
$A \rightarrow a$
$A \rightarrow \varepsilon$
for $A, B \in N$, $a \in Σ$, and $\varepsilon$ the empty string, and

$S ∈ N$ is the start symbol.


Now the question is: Can a regular grammar also be ambiguous?
There do indeed exist ambiguous regular grammars. Take for example

$S\rightarrow A~|~B$

$A\rightarrow a$

$B\rightarrow a$
We know from Chomsky's Heirarchy of Languages that every regular language is also a context free language. 

We also know every regular language is also a regular grammar.

Therefore, every regular grammar is also a context free grammar. Since CFGs can be abmbiguous, therefore by logic, some regular grammars can be ambiguous. (not ALL but there exists, in this case).