Let m 1 be a given modules prove each of the following prop

Let m > 1 be a given modules prove each of the following properties of addition and/or multiplication in Z_n Multiplication is associative; that is, if A, B, C sum Z_n, then (A. B) C = A. (B C) Multiplication is distributive over determine, that is, if A, B, C sum Z_n, then A. (B + C) = (A - B) + (A - C)

Solution

(a)   Let A = [a], B = [b], and C = [c] Z_n, then we have

A (B C) = [a] ([b] [c])

= [a] [bc]

= [a(bc)]

= [(ab)c] [Multiplication of integers is associative]

= [ab][c]

= ([a][b])[c] = (A B) C

Hence multiplication is associative in Z_n.

Also, (A + B) + C = ([a] + [b]) + [c]

= [a + b] + [c] [By definition \"[]\" ]

= [(a + b) + c]

= [a + (b + c)] [Addition of integers is associative]

= [a] + [b + c]

= [a] + ([b] + [c]).= A + (B + C)

Hence addition is associative in Z_n.

(b)   Let A= [a], B = [b], and C = [c] Z_n.

Now, we have

A (B + C) = [a] ([b] + [c])

= [a] [b + c]

= [a (b + c)]

= [ab + ac], [By distributive property of integers]

= [ab] + [ac]

= [a] [b] + [a] [c] = AB + AC

Thus, the distributive property is satisfied in Z_n.