Question

Prove that the set R= {0,1,2,3,4} is a ring with respect to addition and multiplication.​

Answer 1

                                                               

(1) Since all element of the table (1) belongs to the set, that is it is closed under addition.

(2) We know that addition is always associative, so it is associative over addition.

(3) In table (1) 0 is the identity element which is belongs to R.

(4) Additive inverse of 0, 1, 2, 3 and 4 are 0, 4, 3, 2 and 1 respectively.

(5) Since the element is equidistance from the main are equal to each other hence the addition is commutative.

(6) Since all element of the table (2) belongs to the set, that is it is closed under multiplication.

(7) We know that multiplication is always associative, so it is associative over multiplication.

(8) The multiplication is always distributive. (left as well as right)

Hence R = (0, 1, 2, 3, 4) satisfy all the axioms. Hence (R, +, ) is a ring.