Rings and Fields

Field example

$\mathbb{R}$: set of real numbers.

$0, 1 , \in , \mathbb{R}$: two points on a number line.

$x$: any point on the number line, distance from $0$ relative to $1$.

$x_1 + x_2$: addition.

$x_1 \times x_2$: multiplication.

$x_1 \times (x_2 + x_3) = (x_1 \times x_2) + (x_1 \times x_3)$.

$1$: identity.

$x_1 + x_2 = x_2 + x_1$: order doesn’t matter.

$x_1 \times x_2 = x_2 \times x_1$: order doesn’t matter.

$-x$: inverse of $x$ for $+$.

$x^{-1}$: inverse of $x$ for $\times$.

$2 \times \frac{1}{2}$: inverses.

$\mathbb{Z}$: set of integers.

$\mathbb{Z} \subset \mathbb{R}$: integers are a subset of reals.

$\frac{1}{2}$: not an integer.

$2$: has no inverse for $\times$ in $\mathbb{Z}$.

$F(+, \times)$: field.

$F$: a set.

$+, \times$: binary operations.

$F(+)$: commutative group, identity $0$.

$(F \setminus \{ 0 \})(\times)$: commutative group with identity $1 \neq 0$.

$f_1 \times (f_2 + f_3) = (f_1 \times f_2) + (f_1 \times f_3)$.

$(f_1 + f_2) \times f_3 = (f_1 \times f_3) + (f_2 \times f_3)$.

Same as a field except…

$\times$: might not be commutative, might not have inverses.

$r_1, r_2$: rings can have elements where $r_1 \times r_2 = 0$ and neither is $0$.

##∑ Other examples

$\mathbb{Q}$: set of rationals $\{\frac{a}{b} \mid a, b \in \mathbb{Z} \}$.

$\mathbb{C}$: set of complex numbers $\{a + bi \mid a, b \in \mathbb{R} \}$.

$\mathbb{Z}_7$: integers modulo 7.

$\mathbb{R} \times \mathbb{R}$: Cartesian plane.