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1. (10 points) Let R be a commutative ring with identity, and let geR be an idempotent other than 0 or 1. Show that 1-a is also an idempotent, that the ideal gR is also a commutative ring with identity (possibly not the same as that of R), and that R is isomorphic to the direct sum of gR and (1-a)R. [Hint: It may be useful to examine the example with R=Z6 and a=3.] 2. (10 points) Let R be a finite commutative ring with unity. Prove that every prime ideal of R is maximal. Give an example of a finite commutative ring with unity which has more than one prime ideal, and an example with an ideal which is not prime. 3. (10 points) Let R be the ring of 2x2 matrices over the real numbers, and let m be the matrix I: 1] Prove that the set mR is not an ideal of R. Part II: This part corresponds to the material on the second examination. 5. (10 points) Let p be a prime. Compute the number of irreducible polynomials over 3a of the form x³+ax²+bx+c. [Hint: Count the number of reducible polynomials of that form. You should first check how a reducible cubic polynomial can factor; the answer to question 6 on the second test will probably be useful.] 6. (10 points) In the ring Z[V2], which of the numbers 7, 11, and 17 are irreducible? Factor the ones which are not. Part III. This part corresponds to material since the second examination. 7. (10 points) Express (3+v2)-¹ in the form a+bv2, where a and b are rational numbers. 8. (10 points) Let a be a zero of the polynomial x³+x+1 in some extension field of Z2. Find the other two zeroes of that polynomial in Z2{]. (That is, write them in the form a a2+b a +c, where a.b.ceZ2. 9. (10 points) Write out V6 in the 5-adics, out to the 5³ term.

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Abstract Algebra Problems
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