Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2004 / Algebra / 3

A ring is a set with an abelian-group addition, associative multiplication and both distributive laws; here we use rings with multiplicative identity. A field is a nonzero commutative ring in which every nonzero element has a multiplicative inverse.

The positive integers form neither a ring nor a field. They do not contain an additive identity zero, and they also lack additive inverses.

$\mathbb{Z}/4\mathbb{Z}$ is a ring but not a field. Quotient operations inherit associativity and distributivity from the integers, and residues have additive inverses. However the nonzero residue $\overline{2}$ satisfies $\overline{2}\overline{2}=\overline{0}$, so it cannot be invertible. A field has no nonzero zero divisors.

$\mathbb{Z}/3\mathbb{Z}$ is a field, and consequently also a ring. Its nonzero elements are $\overline{1}$ and $\overline{2}$, with inverses $\overline{1}$ and $\overline{2}$ respectively. The quotient ring axioms are inherited as in the preceding part.

This is the Gaussian integer ring $\mathbb{Z}[i]$. It contains zero and one and is closed under addition, additive inverses and multiplication, sinceIts operations inherit the ring axioms from $\mathbb{C}$. It is a ring but not a field: the nonzero element 2 has complex inverse $1/2$, which is not a Gaussian integer.