Let K be a field. Consider the following set -{(::)- a 0 R:= Je K?x2: a1, az € K a2 (Ka,+) (a) Show that (R, +) with the matrix addition as operation has a commutative subgroup of is. (b) Show for all "A, BER: A, BER = A BER (K²x2, +,·), that (R, +, ) with the matrix addition and (c) Using (a).(b) and the ring properties of matrix multiplication as linkage " gen forms a commutative ring with unity. It is important to avoid proving ring properties directly (K²x2,+,'), follows directly from (a).(b) or the ring properties of can be derived (d) Show that (R, +, ) is not a zero divisor. (e) Let (R, ) the unit group of (R, +, ). Show it: A ER A ERA det A #0
Let K be a field. Consider the following set -{(::)- a 0 R:= Je K?x2: a1, az € K a2 (Ka,+) (a) Show that (R, +) with the matrix addition as operation has a commutative subgroup of is. (b) Show for all "A, BER: A, BER = A BER (K²x2, +,·), that (R, +, ) with the matrix addition and (c) Using (a).(b) and the ring properties of matrix multiplication as linkage " gen forms a commutative ring with unity. It is important to avoid proving ring properties directly (K²x2,+,'), follows directly from (a).(b) or the ring properties of can be derived (d) Show that (R, +, ) is not a zero divisor. (e) Let (R, ) the unit group of (R, +, ). Show it: A ER A ERA det A #0
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.3: The Characteristic Of A Ring
Problem 24E
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plz provide answer for part d and e
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