I'm readingA concrete introduction to higher algebraby Lindsay Childs, and there's a theorem saying "If $\displaystyle R$ is a commutative ring with identity, then $\displaystyle M_n(R)$ is a ring with identity". I wonder if $\displaystyle R$ really has to be commutative, i.e. has a commutative multiplication (since addition in any ring is commutative by definition). I presume that distributivity, associativity of multiplication and commutativity of addition in $\displaystyle M_n(R)$ depend only on commutativity of addition in $\displaystyle R$. Is it the case that if multiplication in $\displaystyle R$ fails to be commutative, $\displaystyle M_n(R)$ wouldn't be a ring with identity?