Let R be a ring with 10. A nonzero element a is called a left zero divisor in R if there is a nonzero element x = R such that ax = 0. Symmetrically, b 0 is a right zero divisor if there is a nonzero y € R such that yb= 0 (so a zero divisor is an element which is either a left or a right zero divisor). An element u € R has a left inverse in R if there is some SER such that su = 1. Symmetrically, v has a right inverse if vt = 1 for some 1 € R. (a) Prove that u is a unit if and only if it has both a right and a left inverse (i.e., u must have a two-sided inverse). (b) Prove that if u has a right inverse then u is not a right zero divisor. (c) Prove that if u has more than one right inverse then u is a left zero divisor.
Let R be a ring with 10. A nonzero element a is called a left zero divisor in R if there is a nonzero element x = R such that ax = 0. Symmetrically, b 0 is a right zero divisor if there is a nonzero y € R such that yb= 0 (so a zero divisor is an element which is either a left or a right zero divisor). An element u € R has a left inverse in R if there is some SER such that su = 1. Symmetrically, v has a right inverse if vt = 1 for some 1 € R. (a) Prove that u is a unit if and only if it has both a right and a left inverse (i.e., u must have a two-sided inverse). (b) Prove that if u has a right inverse then u is not a right zero divisor. (c) Prove that if u has more than one right inverse then u is a left zero divisor.
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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Transcribed Image Text:Let R be a ring with 10. A nonzero element a is called a left zero divisor in R if there is
a nonzero element x = R such that ax = 0. Symmetrically, b 0 is a right zero divisor if
there is a nonzero y € R such that yb= 0 (so a zero divisor is an element which is either
a left or a right zero divisor). An element u € R has a left inverse in R if there is some
SER such that su = 1. Symmetrically, v has a right inverse if vt = 1 for some 1 € R.
(a) Prove that u is a unit if and only if it has both a right and a left inverse (i.e., u must
have a two-sided inverse).
(b) Prove that if u has a right inverse then u is not a right zero divisor.
(c) Prove that if u has more than one right inverse then u is a left zero divisor.
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