Is the class of non-regular languages closed against Kleene star?

How to prove that if a language A is not regular then A* isn't regular either?

I have tried the usual methods with no result.

• What makes you think that this claim is correct? – Shaull Nov 4 '13 at 11:01
• If you don't find a proof, look for a counter-example... – Denis Nov 4 '13 at 11:09
• Hint: try a non-regular language over a unary alphabet $\{a\}$. See where that gets you. – Shaull Nov 4 '13 at 11:30
• For Kleene star for a language over unary alphabet, see here. – Hendrik Jan Nov 4 '13 at 13:38
• Actually, you don't even need to consider languages over a unary alphabet. Consider, for instance, non-regular languages containing at least all strings of length one over any alphabet. – Patrick87 Nov 4 '13 at 14:45

Hint: Suppose $L$ is any language over the alphabet $\Sigma$. If $L$ is not regular then so is $L+\Sigma$, yet $(L+\Sigma)^* = \Sigma^*$ is regular.