# Algebra2010Aug - Algebra Part of Qualifying Examination...

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Unformatted text preview: Algebra Part of Qualifying Examination, August 23, 2010 Instructions: Do all questions, justify your answers with the necessary proofs. All rings are associative (not necessarily commutative) with identity, and all modules are left unitary modules. We denote by Z the ring of integers, and by R , Q the fields of real and rational numbers, respectively. 1. Given a ring R, the opposite ring R op has the same underlying abelian group as R and a new multiplication ◦ defined by a ◦ b = ba, for all a,b ∈ R, where juxstaposition denotes the original multiplication in R. Let M m,n ( R ) be the set of m × n matrices with entries from R. The set M n ( R ) = M n,n ( R ) is a ring with respect to the usual addition and multiplication of matrices. The sets M n ( R ) and M n ( R op ) coincide as abelian groups, but have different ring structures. For all A ∈ M m,n ( R ) ,a ∈ R, we write a ◦ A = Aa and A ◦ a = aA for the scalar multiplication. For all B ∈ M n,p ( R ) , we write AB for the product of A and B over R, and A ◦ B for their product over...
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Algebra2010Aug - Algebra Part of Qualifying Examination...

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