MTH5101 Ring Theory: Guide to Coursework 4A Note: These are by no means complete solutions! 1. (a) Write down the definition of an integral domain. Give an example of an integral domain, and an example of a domain that is not an integral domain. An integral domain is a domain containing no zero-divisors, that is, a domain such that if a · b = 0 then either a = 0 or b = 0. An example of an integral domain is the ring Z, and an example of a domain that is not an integral domain is Z=6Z (as [2]6 · [3]6 = [0]6). 2. Consider the domain R = Z=14Z = f[0]14; [1]14;:::; [13]14g. (a) List all the zero-divisors of R, and explain why they are zero-divisors. The zero-divisors correspond to the integers that are not coprime to 14: [2]14; [4]14; [6]14; [7]14; [8]14; [10]14; [12]14: You should check that each of them can be multiplied by another (pos- sibly more than one) non-zero element to get [0]14. (b) List all the units of R, and their corresponding inverses. The units correspond to the integers that are coprime to 14: [1]14; [3]14; [5]14; [9]14; [11]14; [13]14: Again, make sure to know for each of them which is the (unique) element they need to be multiplied by to get [1]14. (c) Is R an integral domain? Is R a field? The domain R has zero-divisors, so it is not an integral domain. It is also not a field, because there are non-zero elements that are not units. 3. Let R = Z=4Z, and consider the ring R[x] of polynomials with coefficients in R. (a) Is R[x] a domain? Explain. Since R is a domain, R[x] is also a domain (a commutative ring with identity). For example, what is the identity of R[x]? (b) Is R[x] an integral domain? Explain. The ring R has zero-divisors, and this leads to many zero-divisors in R[x]. For example, ([2]4 x) · ([2]4 + [2]4 x) = 0 (the zero polynomial), but none of the factors is the zero polynomial. 2 (c) Is the element [2]4 + [0]4 x + [2]4 x 2 R[x] a zero-divisor? It is. Can you find another non-zero polynomial which multiplied by it is equal to 0? (d) Is the element [2]4 + [1]4 x 2 R[x] a zero-divisor? It is not. To see this, think about what the leading coefficient (the coefficient with the highest power of x) is after multiplying by another polynomial. 2 (e) Is the element [2]4 + [3]4 x + [2]4 x 2 R[x] a unit? It is not. Think about the constant coefficient after multiplying by another polynomial. (f) Is the element [3]4 + [2]4 x 2 R[x] a unit? It is! Can you find its inverse?.