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Let K be a field. Consider the following set {(::)- a1 0 R:= a2 a1 (a) Show that (R, +) with the matrix addition as operation has a commutative subgroup of (K*, is. (b) Show for all " A, BER: A, BER = A.BER

Let K be a field. Consider the following set {(::)- a1 0 R:= a2 a1 (a) Show that (R, +) with the matrix addition as operation has a commutative subgroup of (K*, is. (b) Show for all " A, BER: A, BER = A.BER

Elements Of Modern Algebra
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 b

Let K be a field. Consider the following set {(:) a1 0 R:= E K²x2 : a2 a1 (a) Show that (R, +) with the matrix addition as operation has a commutative subgroup of (K2 is. (b) Show for all "A, BER: A, BER = A · BER (K²x2,+,·), (c) Using (a).(b) and the ring properties of that (R, +, ) with the matrix addition and 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 E R + A E RA det A +0
Transcribed Image Text:Let K be a field. Consider the following set {(:) a1 0 R:= E K²x2 : a2 a1 (a) Show that (R, +) with the matrix addition as operation has a commutative subgroup of (K2 is. (b) Show for all "A, BER: A, BER = A · BER (K²x2,+,·), (c) Using (a).(b) and the ring properties of that (R, +, ) with the matrix addition and 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 E R + A E RA det A +0
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