I am reading an article. There is a step in which I suspect that they use a "result" that "Let $A$ be a local artinian $k$-algebra with residue field $k$. If $A$ is regular then $A$ is nothing but $k$." 


I do not know whether it is true or not . If it is true, can anyone give me a source including it .


Thank you in advance .



Regular local artinian k-algebra with residue field k is k
A local ring $(A,\mathfrak{m})$ is regular iff the minimal number of generators of $\mathfrak{m}$ equals the Krull dimension of $A$. But Artinian rings have Krull dimension $0$, i.e. $\mathfrak{m}=0$, hence $A=k$. The latter fact can for example be found in Atiyah and MacDonald's book on Commutative Algebra.


