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By Garrett Birkhoff

This vintage, written through younger teachers who grew to become giants of their box, has formed the knowledge of recent algebra for generations of mathematicians and is still a beneficial reference and textual content for self learn and faculty classes.

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Show that • N is a subring of R, • N is non-commutative if n � 3 and n = 0, V :z: E N . • :z: 9. lg R, show that ab is nilpotent if and only if ba is nilpotent. Can one say the same for zero-divisors? 10. Let R = M2 (Z). Give examples of matrices in R having the following properties. • A E R is such that A is a zero-divisor but not nilpotent and • A, B E R are such that both A and B are nilpotent but A + B is not nilpotent . Verify that for such a pair AB f. mBA, V m E :Z . 1 1 . e . , a b any p ositive integer n , prove the binomial expansion that = b a .

I = ( x ) t for some E I. 5 Right ideal generated by X: The right ideal generated b11 X is the smallest right ideal in R containing X, or equiv­ alently, it can be defined as the intersection of all the right ideals in R containing X which can be seen to be equal to finite finite n; E 71: z; E X r; E R 1/j E X { L nixi + L y; r; } In particular, 1 . If X = 0, the right ideal generated by 0 is (0) . 2 . If X = {x}, the right ideal generated by x is {nx + x r I n E 7l. , r E R}, ( = {x8 I 8 E R} , if 1 E R) .

RINGS several variables over a ring R are defined in a similar way and are respectively denoted by R[Xf\ , x: 1 J and R < X1 7 • • · , Xn > . 11 · · Opposite Rings 1 . 1 1 . 1 Opp osite ring: Given a ring R, let R0, or R0P (read as R­ opposite) , be the same set R. Define addition ( +) and multiplication ( * ) on R0 as a + b = a + b and a * b = b a, V a, b E R. 5ite ring of R. It is clear that (R0) 0 = R. · 1 . 1 1 . 2 Remarks: 1 . R = R0 if and only if R is commutative. 2 . R has unity if and only if � has unity.

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