Question

8. let salle &]: xy, 2 e R} a). Prove that (5, +,-) is a ring, where t' and are the usual addition and multiplication of matrices. (You may assume standard properities of matrix Operations ) b). Let T be the set of matrices in 5 of the form { x so]. Prove that I is an ideal in the ring s. c). Let & be the function f: 5-71R, given by f[ 8 ] = 2 i prove that f is a ring homomorphism GIJ Explain why kerf = T, using the definition of kernel. d). Determine whether or not the quotient ring SIT is isomorphic to the ring R. Justify fully. your claim

Answer 1

Answer 2

8. 수 x g: x,y, 2 ER - 0 2 Ring. (a) 1 am We have to prove that, S is a Then we need to prove the following - Ido (S, +) be abelian Group s has the closure property and Associative propesty. I B=La s GM 212 Ss, R1 gi 2 . [ 22 Kel A = 2 A, BE S. $1 1-1 Az 22 A+ B = 0 2 Iz ty Rp the % +M2 = Fre+291 2212 Z+22 s y Na 11 12 = Br .21 0 ($,+) is Commutative . [ . is the additive identity element 그 and is the additive inverse [ i ܕ݁ܳ ܒܪܢ of pa y 0 2 Hence, (S,+) is an Abelian Group. CS Scanned with CamScanner

Since, Hence (s: ja ?i (S,') is monoid. se Manz , by Heri detary property S satisfies closure and Associative property: 7 is the multiplicative identity of (s) :) a monoid. 3. Distributive Property. co element Siti xet A, B, C Es nez M3 23 . As (02]. -- A: (8-0)-[23:1 21 o 1 y, H2 + 163 ZAM 2+23- I seeing the 13 x 9 + 24 - Y3 +7, 222.53 2-22+2.23 Hy % ☺ B 73 Zz 0 1.8426 DN 36:10 Pred the Xz A tej. Mz +7 27 74.23 2; Z2 +2 23 Ö CS Scanned with CamScanner

(6) Now , Hence, (S, + b) a Ring 12(4):e9er? ERT ļ We have to prove that I is an ideal of si have to prove the the following properties L. (T,+) is a sub subgroup of a (s, t) Son we og det net A, BET A, B t T sit. M2 A = [X2 B5 ( 7 se ] siz Ya 82 si f A+BE really sty 2 ☺ nez 19 BET Heriditary FLET Lo T is closed. (h The Associative by property Tes, [oo] et sT contains the identity element, (N) [ x 37 the inresse element of [n Hence; (Telis a subgrong of (s, t) is 0 nyl o CS Scanned with CamScanner

the 2. for every te T andres produd tx ET het t= 1 ng [ 12 por ] [ Mig 22 ses y tin= PE mix sa q +9, ZZ U o rez 137 ET 0 of S. the T is an Hence, an ideal ring (c) fis > R sit (i) Ket x y 2, C f([O 2 ' y sit. A = Ket A, B Es ner 2 B = Leo 22 ng D2 $(AB) =f +( ko y 22 sa l x g +7, 27 21 22 2.22 DAT CS Scanned with CamScanner

22 f(49:4(B) =f F (D3) 2.22 . 'J homomorphism. +(4.B) f(49. f16). Hence fisa ker f = {lo y :+(-1)-07 (@) f 2-0 in y xyy y yet? 0 Кел worf = { [" :f T E CS Scanned with CamScanner