rings and fields subject

Exercise No. 6. 1. Let R be a ring with unity 1 and x = (0, 1,0, 0, 0,.....]. Then calculate x2, x3, x*,x5, x* by doing the complete steps. 2. Verify whether or not Z, [x] is an integral domain. 3. Given 2x* + x3 + 3x + 1, and 4x* + 2x2 + 3x + 3, x* + 3x + 2x'+x+26Z,[x]. Find their additive inverses in Z, [x]. 4. If 5x3+ 4x' + x+ 4, and 3x* + 5x3 + 4x + 1, x* + 5x3 + 4x- + 4x + 56Z,[x]. Then verify the associative law of "+" for these elements in Zo[x]. 5. If 2x2 + 3x + 4, and 2x2 + 3x + 1, 24x' + 4x + 16 Z, [x]. Then verify the associative law of "." for these elements in Z,[x]