Exercise 5.1.4 Verify that if then The following theorem establishes some of the
ID: 3282266 • Letter: E
Question
Exercise 5.1.4 Verify that if then The following theorem establishes some of the properties of the two operations, +,. According to these, (Z, + is a (commutative) ring with zero the class R(0,0)), and identity the class RI(1,0) Theorem 5.1.3 Let z, y,2 E Z. The follouing hold true for (Z,+,,): (1) The addition is commutative, associative, has a neutral element, and every element has an opposite: (V)(R(0,0)- ()Br - Ri(0,0)) (2) The multiplication is commutative, associative, and has a neutral element Va R(1,0)) (3) The multiplication distributes over the additionExplanation / Answer
Given R[(n, m )]= R[(n1, m1 )] and R[(n', m' )] = R[(n1', n2')] then
LHS = R[(nn'+ mm', nm'+ n'm )]
= R[( mm'+ nn', nm'+ n'm )] { Since x+y= y+x }
= R[( mm'+nn', nm'+mn' )] { Since x.y= y.x }
= R[( mm',nm') + (nn', mn')]
= R[( m, n)m' +(n, m) n']
= R[( m, n) m' +(m,n) n']
= R[(m,n)(m' + n' )]
= R[(m1, n1)(m1'+n1')] { Since xR[(1,0)] = x}
= R[( m1, n1)m1' + (n1, m1 ) n1']
= R[(m1m1' , n1m1') + (n1n1' , m1n1')]
= R[(m1m1'+n1n1' , n1m1'+ m1n1')]