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# h9 - 1 Homework IX 3.9 Find all the units in the...

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1 Homework IX: July 31, 2009 3 . 9 . Find all the units in the commutative ring R = F ( R ) in Example 3.11(i). We claim that f is a unit in R if and only if f ( r ) = 0 for all r R . If f is a unit, then there is g R with fg = 1; that is, ( fg )( r ) = f ( r ) g ( r ) = 1 for all r R . Hence, f ( r ) = 0. Conversely, suppose that f ( r ) = 0 for all r R . Define g by g ( r ) = ( 1 /f ( r ) ) for each r . Then ( fg )( r ) = f ( r ) g ( r ) = 1 for all r ; hence, fg = 1, and so f is a unit. 3 . 19 . Define F 4 to be the set of all 2 × 2 matrices a b b a + b , where a, b I 2 . ( i ) . Prove that F 4 is a commutative ring whose operations are matrix addition and matrix multiplication. Clearly, the identity matrix I lies in F 4 . F 4 is closed under matrix addition: a b b a + b + c d d c + d = a + c b + d b + d a + c + b + d ; F 4 is closed under matrix multiplication. a b b a + b c d d c + d = ac + bd ad + bc + bd ac + bd + bd bd +( a + b )( c + d ) . This formula also shows that F 4 is commutative. All the other properties we need: associativity, distributivity, etc. hold because F 4 is a subring of ring of all 2 × 2 matrices with entries in I 4 . I will allow you to refer to the usual properties

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h9 - 1 Homework IX 3.9 Find all the units in the...

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