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.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.4: Maximal Ideals (optional)
Problem 26E: . a. Let, and . Show that and are only ideals of
and hence is a maximal ideal.
b. Show...
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