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 will have no identity.
I'm reading A concrete introduction to higher algebra by Lindsay Childs, and there's a theorem saying "If is a commutative ring with identity, then is a ring with identity". I wonder if 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 depend only on commutativity of addition in . Is it the case that if multiplication in fails to be commutative, wouldn't be a ring with identity?