(2) Simplifying each expression in the given ring

R

.\ (a)\

R=M_(3)(Z_(5))\ [[1,2,4,3,2\ 1,1,3]]\ 1\ 0+[[2,1,3

]]

\ 1,1,1\

[

2,2,3]\ [1,2,4

]

\ 0,3,2\

[[

1,1,3]][[2,1,3,1,1,2,3]]\ 2\ 1

\ (b)\

R=Z_(4)[x]\ (3x^(5)-x^(4)+x^(2)-1)+(x^(2)-x+2)\ (3x^(5)-x^(4)+x^(2)-1)(x^(2)-x+2)

\ (c)\

R=M_(2)(Z[x])\ [[1+x,1,x^(2)]]\ x^(3)+[[1+x,x^(2)

]]

\

[

1+x,1-x]\ [1+x,1

]

\

[[

x^(3),x^(2)]][[1+x,x^(2),1-x]]\ 1+x

\ (d)\

R=M_(2)(Z_(2)[x])\ [[1+x,1,x^(2)]]\ x^(3)+[[1+x,x^(2)

]]

\

[

1+x,1-x]\ [1+x,1

]

\

[[

x^(3),x^(2)]][[1+x,x^(2),1-x]]\ 1+x

\ (e)\

R=Z_(2)[G] where G={1,t,t^(2),t^(3),t^(4):t^(5)=1}\ (1+t+t^(2)+t^(3))+(t+t^(2)+t^(4))\ (1+t+t^(2)+t^(3))(t+t^(2)+t^(4))

(2) Simplifying each expression in the given ring R. (a) R=M3(Z5)101231423+212112313101231423212112313 (b) R=Z4[x](3x5x4+x21)+(x2x+2)(3x5x4+x21)(x2x+2) (c) R=M2(Z[x] (d) R=M2(Z2[x])[1+xx31x2]+[1+x1+xx21x][1+xx31x2][1+x1+xx21x] (e) R=Z2[G]whereG={1,t,t2,t3,t4:t5=1}.(1+t+t2+t3)+(t+t2+t4)(1+t+t2+t3)(t+t2+t4)