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

  1. #1
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    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?
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  2. #2
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    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.
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