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ProblemSet9

# ProblemSet9 - it has nontrivial right ideals(b Let R be the...

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Homework 9 Due: Monday, April 12, 2010 Problems after the double bar will not be collected but it is recommended that you do them. My assumption is that you will do them. 1. Show that there is an inﬁnite number of solutions to x 2 = - 1 in the ring of quaternions { α 0 + α 1 i + α 2 j + α 3 k | α i R } . 2. Show that ±² a b - b a ³ ´ ´ ´ a,b R µ is a ﬁeld. 3. If R is a ﬁnite integral domain, show that R is a ﬁeld. (Do not assume that all integral domains contain an identity. Although some books add that assumption to the deﬁnition, we did not.) 4. (a) Show that the ring of 2 × 2 matrices over the reals has nontrivial left ideals. (Similarly,
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Unformatted text preview: it has nontrivial right ideals.) (b) Let R be the ring of 2 × 2 matrices over the reals and suppose that I is an ideal of R . Show that I = (0) or I = R . 5. Let R = Z [ i ], the ring of Gaussian integers . Starting with R , construct a ﬁeld having 49 elements. 6. Find all the units (invertible elements) in the ring Z 24 7. Let R be a ring in which x 4 = x for every x ∈ R . Prove that R is commutative. 8. If a,b ∈ Z such that either 3 6 | a or 3 6 | b (or both) show that 3 6 | ( a 2 + b 2 )...
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