5 Groups, Rings, Fields[1ZD]

We review these definitions.

We assume that $0\ne 1$ (otherwise $\{0\}$ would be a ring).

Examples of commutative rings are: integer numbers $\mathbb{Z}$, polynomials $A[x]$ with coefficients in a commutative ring $A$.

An example of a non-commutative ring is given by matrixes ${ℝ}^{n\times n}$, with the usual operation of multiplication and addition.

Some field examples are: rational numbers $\mathbb{Q}$, the real numbers $ℝ$ and the complex numbers $ℂ$.

Examples of ordered field are: rational numbers $\mathbb{Q}$ the real numbers $ℝ$. The complex numbers $ℂ$ do not allow an ordering satisfying the above properties (see exercise [08V]↺↻).

¹

E73

[1ZM]↺↻

E73

[1ZP]↺↻

E73

[29C]↺↻

E73

[1ZR]↺↻

E73

[1ZS]↺↻

E73

[203]↺↻

E73

[1ZT]↺↻

E73

[1ZV]↺↻

E73

[1ZX]↺↻

E73

[1ZY]↺↻

E73

[1ZZ]↺↻

E73

[08V]↺↻

E73

[200]↺↻

E73

[202]↺↻

E73

[20T]↺↻

E73

[205]↺↻