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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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