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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, b0 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 t € R. Prove that if R is a finite ring then every element that has a right inverse is a unit (i.e., has a two-sided inverse).

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, b0 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 t € R. Prove that if R is a finite ring then every element that has a right inverse is a unit (i.e., has a two-sided inverse).

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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Please provide explanation with the taken steps, I know something about surjective and injective being used in the answer but thats as far as I can get, Im quite new to ring theory. Thank you in advance

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. Prove that if R is a finite ring then every element that has a right inverse is a unit (i.e., has a two-sided inverse).
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. Prove that if R is a finite ring then every element that has a right inverse is a unit (i.e., has a two-sided inverse).
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