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Let $A$ be an $n$ by $n$ matrix over a field $K$, define $R:=\{ \sum_{i=0}^{\infty} c_i A^i$ with only finitely many $c_i \neq 0 \}$

Could anyone help me show you can turn $R$ into a commutative ring with $1$?

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    @LHS Man you got me confused too!2012-05-08

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Let $A$ be some $n\times n$ matrix over a field $K$ and consider $R := \{ \sum_{i=0}^{\infty} c_i A_i\}$ where all but finitely many $c_i$ non-zero. Now the identity matrix is in here because it can be written as $1 \times A^0$. Therefore it follows that the identity matrix is the multiplicative identity of the ring $R$.

It is easy to see from here that $R$ can be made into a unital commmutative ring.