## Transcribed Text

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

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.