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12. (a) Let R be a ring without zero divisors. Show that if ab = 1 then ba = 1 as well. Thus, a and b are units of R. (b) Show that if a, b, c ER with ab = 1 and ca = 1, then b = c, and thus a (and b) are units. Conclude that if ab = 1 but ba 1, then neither a nor bare units.

12. (a) Let R be a ring without zero divisors. Show that if ab = 1 then ba = 1 as well. Thus, a and b are units of R. (b) Show that if a, b, c ER with ab = 1 and ca = 1, then b = c, and thus a (and b) are units. Conclude that if ab = 1 but ba 1, then neither a nor bare units.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I know according to guidedlines u will do a part only But please I request u do all parts Please and please
12. (a) Let R be a ring without zero divisors. Show that if ab = 1 then ba = 1 as well. Thus, a and b are units of R. (b) Show that if a, b, c ER with ab = 1 and ca = 1, then b = c, and thus a (and b) are units. Conclude that if ab = 1 but ba 1, then neither a nor b are units. (c) Let F be a field and FIX] the polynomial ring with coefficients from F. FIX is an F-vector space in a natural way, so by Exercise 11, R = Endr(FIX]) is a ring. Give an example of a, b e R with ab = 1 but ba # 1.
Transcribed Image Text:12. (a) Let R be a ring without zero divisors. Show that if ab = 1 then ba = 1 as well. Thus, a and b are units of R. (b) Show that if a, b, c ER with ab = 1 and ca = 1, then b = c, and thus a (and b) are units. Conclude that if ab = 1 but ba 1, then neither a nor b are units. (c) Let F be a field and FIX] the polynomial ring with coefficients from F. FIX is an F-vector space in a natural way, so by Exercise 11, R = Endr(FIX]) is a ring. Give an example of a, b e R with ab = 1 but ba # 1.
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