MTH 461 University of Miami Modern Algebra Questions

**MTH 461 University of Miami Modern Algebra Questions**

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MTH 461: Survey of Modern Algebra, Spring 2021 Homework 9 Homework 9 1. Determine which of the following subsets are ideals. For example, Q Ă R means: is Q an ideal in the ring R? Justify your answer in each case. (a) 2Z Ă Z (b) Q Ă R ? ? (c) ta ` b ´5 ∶ a ` b ” 0 pmod 6qu Ă Zr ´5s (d) R Ă Rrxs (e) tf px, yq P Rrx, ys ∶ f p4, 5q “ 0u ? 2. Consider the ideal in the ring Zr ´5s given by ? ? ? I “ p3, 1 ` ´5q “ t3x ` p1 ` ´5qy ∶ x, y P Zr ´5su ? ? ? Suppose I is principal and generated by some a`b ´5 P Zr ´5s, i.e. I “ pa`b ´5q. ? ? (a) Since 3 P I, we have 3 “ pa ` b ´5qpc ` d ´5q for some c, d P Z. Find all possi? bilities for a ` b ´5 based on this relation. (b) Do the same as in (a) but using 1 ` ? ´5 P I instead of 3 P I. (c) Using your results in (a) and (b) show that I cannot be a principal ideal. 3. Let R be a commutative ring

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MTH 461 University of Miami Modern Algebra Questions

5 attachmentsSlide 1 of 5

### UNFORMATTED ATTACHMENT PREVIEW

MTH 461: Survey of Modern Algebra, Spring 2021 Homework 9 Homework 9 1. Determine which of the following subsets are ideals. For example, Q Ă R means: is Q an ideal in the ring R? Justify your answer in each case. (a) 2Z Ă Z (b) Q Ă R ? ? (c) ta ` b ´5 ∶ a ` b ” 0 pmod 6qu Ă Zr ´5s (d) R Ă Rrxs (e) tf px, yq P Rrx, ys ∶ f p4, 5q “ 0u ? 2. Consider the ideal in the ring Zr ´5s given by ? ? ? I “ p3, 1 ` ´5q “ t3x ` p1 ` ´5qy ∶ x, y P Zr ´5su ? ? ? Suppose I is principal and generated by some a`b ´5 P Zr ´5s, i.e. I “ pa`b ´5q. ? ? (a) Since 3 P I, we have 3 “ pa ` b ´5qpc ` d ´5q for some c, d P Z. Find all possi? bilities for a ` b ´5 based on this relation. (b) Do the same as in (a) but using 1 ` ? ´5 P I instead of 3 P I. (c) Using your results in (a) and (b) show that I cannot be a principal ideal. 3. Let R be a commutative ring

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