Let $\Bbb K$ be a field. Can we say that $\Bbb K^n$ is a $\Bbb K$-algebra?
I think it's a $\Bbb K$-algebra. Because first of all $\Bbb K^n$ is a ring with respect to component wise addition and multiplication. Also $\Bbb K^n$ is a $\Bbb K$-vector space with respect to component wise scalar multiplication. Moreover the scalar multiplication in $\Bbb K^n$ is compatible with the ring addition and multiplication endowed with $\Bbb K^n$ as a ring. Therefore in my opinion $\Bbb K^n$ has to be a $\Bbb K$-algebra. Am I correct in my argument?
Any valuable suggestions regarding this will be highly appreciated. Thank you very much.