# Math Help - Linear Algebra: please clarify the statement of theorem

1. ## Linear Algebra: please clarify the statement of theorem

I'm reading A concrete introduction to higher algebra by Lindsay Childs, and there's a theorem saying "If $R$ is a commutative ring with identity, then $M_n(R)$ is a ring with identity". I wonder if $R$ 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 $M_n(R)$ depend only on commutativity of addition in $R$. Is it the case that if multiplication in $R$ fails to be commutative, $M_n(R)$ wouldn't be a ring with identity?

2. No, you can still make that set into a ring because there are matrix rings over noncommutative rings. But if R has no identity then $M_n(R)$ will have no identity.