1. Let R1 and R2 be two rings. Find the units of R1 × R2. 2....

1. Let R1 and R2 be two rings. Find the units of R1 × R2. 2. For what integers N is the ring Z/NZ a field? For what integers is it an integral domain? Give proofs of your assertions. 3. Find all subrings of Z/NZ. 4. Use the Euclidean algorithm to find the GCD d of 780 and 924 and then use your work to find integers r and s such that d = 780r + 924s. 5. Let f(x) = 3x 3 − 4x 2 + 7x − 1 and g(x) = x − 2 be two polynomials in Q[x]. Compute the quotient and remainder when f(x) is divided by g(x). 6. Let R be a ring. An element a ∈ R is called a zero divisor if there exists a nonzero element b in R such that ab = 0. (a) Do zero divisors form a subring of R? Can an element a of R be both a zero-divisor and a unit? (b) Give an example of a ring in which every non-zero element is either a unit or a zero-divisor and in which there are zero divisors other than 0. (c) Give an example of a ring in which some elements are neither units nor zero divisors. 7. Determine which of the following is a ring and if it is a ring, determine if it is a field. Give proofs of your assertions : (a) The set of all purely imaginary numbers : {ri|r ∈ R, i = √ −1}, with the usual operations. (b) The non-negative integers, with usual operations. (c) The set of continupus functions f : R → R where f(0) = 1, with the usual operations of function addition and multiplication. (d) {a + b √ −5|a, b ∈ Q} (⊂ C) with usual addition and multiplication