Answer to Question #149665 in Abstract Algebra for Sourav Mondal

Question

Consider the ring S = R[x] / (x²— 3x + 2).

(i) Give two distinct elements of S with justification.

(ii) Does S have zero divisors ? Justify your answer.

Answer 1

(i) All non-equal polinomials of degree 1 or less are distinct in S. Indeed, their difference is also a polinomial of degree 1 or less. If it equals to 0 in S then it must be divisible by the polynomial x²-3x+2 and its degree must be 2 or more (if it is not zero). The contradiction proves the statement. For example, 1 and x are distinct elements in S.

(ii) According to the above, the polinomials x-1, x-2 and 0 are all distinct in S. As (x-1)(x-2) = x²-3x+2 = 0 in S (t.e. the product of two non-zero elements is equal to 0), therefore, S have zero divisors.