College Algebra Exam Review 259

College Algebra Exam Review 259 - 6.1. A RECOLLECTION OF...

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6.1. A RECOLLECTION OF RINGS 269 Exercises 6.1 6.1.1. Show that if a ring R has a multiplicative identity, then the mul- tiplicative identity is unique. Show that if an element r 2 R has a left multiplicative inverse r 0 and a right multiplicative inverse r 00 , then r 0 D r 00 . 6.1.2. Verify that RŒx 1 ;:::;x n Ł is a ring for any commutative ring R with multiplicative identity element. 6.1.3. Consider the set of infinite-by-infinite matrices with real entries that have only finitely many nonzero entries. (Such a matrix has entries a ij , where i and j are natural numbers. For each such matrix, there is a natural number n such that a ij D 0 if i ± n or j ± n .) Show that the set of such matrices is a ring without identity element. 6.1.4. Show that (a) the set of upper triangular matrices and (b) the set of upper triangular matrices with zero entries on the diagonal are both sub- rings of the ring of all n -by- n matrices with real coefficients. The second example is a ring without multiplicative identity. 6.1.5.
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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