Question: Let R be a ring. An element a: E R is nilpotent if cc = 0 for some positive integer n. (a) Assume that R

Let R be a ring. An element a: E R is nilpotent if cc\" = 0 for some positive integer n. (a) Assume that R is a commutative ring and that 11:, y E R are nilpotents; prove that x + y is nilpotent. (b) To show the necessity of the commutative assumption in part (a): nd nilpotent 2 x 2 real matrices A, B such that A + B is not nilpotent. (Show that your choices for A, B are nilpotent and that A + B is not.) (Hint: you can even nd A, B with only zeros and ones in the entries.) (Hint 2: can a nilpotent matrix be full rank?)

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