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(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. (d) Prove that if R is a finite-dimensional algebra over a field then every element that has a right inverse is a unit (i.e., has a two-sided inverse).1

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