Homework8 - r ∈ R | r is nilpotent(a Prove that Nil R is...

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Homework 8 Math 332, Spring 2010 These problems must be written up in L A T E X, and are due this Thursday, May 13. 1. Consider the ring M 2 ( Z 2 ) = a b c d : a, b, c, d Z 2 . (a) This ring has 9 zero divisors. List them. (b) The ring has 6 units. List them. (c) The units of M 2 ( Z 2 ) form a group under multiplication. Determine the isomor- phism type of this group. 2. Let R be a commutative ring. Recall that an element r R is nilpotent if there exists an n N for which r n = 0. The nilradical of R is the set Nil( R ) = { r
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Unformatted text preview: { r ∈ R | r is nilpotent } . (a) Prove that Nil( R ) is a subring of R . (b) Prove that Nil( R ) is an ideal of R . (c) Prove that every prime ideal of R contains Nil( R ). (d) Prove that the quotient ring R/ Nil( R ) has no nonzero nilpotent elements. 3. An abelian group of order 8192 has elements of the following orders: order 1 2 4 8 16 32 # of elements 1 31 224 1792 2048 4096 Determine the isomorphism type of the group....
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