ABSTRACT
6301. Which of the following sets of numbers form a ring with respect to the usual operations of addition and multiplication:
the set ℤ;
the set nℤ (n > 1);
the set of all non-negative integers;
the set ℚ;
the set of rational numbers where the denominators divide a fixed integer n ∈ ℕ;
the set of rational numbers where the denominators are not divisible by a fixed prime p;
the set of rational numbers where the denominators are powers of a fixed prime p;
the set of real numbers of the form x + y 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756454/84bc859e-388a-4821-81f7-7fb447dc7024/content/eq1818.tif"/> , where x, y ∈ ℚ;
the set of real numbers of the form x + y 2 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756454/84bc859e-388a-4821-81f7-7fb447dc7024/content/eq1819.tif"/> , where x, y ∈ ℚ;
the set of real numbers of the form x + y 2 3 + z 4 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756454/84bc859e-388a-4821-81f7-7fb447dc7024/content/eq1820.tif"/> , where x, y, z ∈ ℚ;
the set of complex numbers of the form x + yi, where x, y ∈ ℤ;
the set of complex numbers of the form x + yi, where x, y ∈ ℚ;
the set of all possible sums of the form a 1 z 1 + a 2 z 2 +…+ anzn , where a 1,a 2,…,an are real numbers, z 1,z 2,…zn are complex roots of 1;
248the set of complex numbers of the form x + y D 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756454/84bc859e-388a-4821-81f7-7fb447dc7024/content/eq1821.tif"/> , where D is a fixed square-free integer, (i.e. not divisible by the square of a prime), and x and y are integers of the same parity?
