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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F244c6516-c7e6-4ded-b620-989e67bc99ea%2F1cee2843-687c-4297-b8d2-facfbd4d36da%2Fru37bq6_processed.png&w=3840&q=75)
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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