5 Groups, Rings, Fields[1ZD]

We review these definitions.

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

Examples of commutative rings are: integer numbers \(ℤ\), polynomials \(A[x]\) with coefficients in a commutative ring \(A\).

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

Some field examples are: rational numbers \(ℚ\), the real numbers \(ℝ\) and the complex numbers \(ℂ\).

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

¹

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[1ZM]↺↻

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[1ZP]↺↻

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[29C]↺↻

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[1ZR]↺↻

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[1ZS]↺↻

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[203]↺↻

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[1ZT]↺↻

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[1ZV]↺↻

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[1ZX]↺↻

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[1ZY]↺↻

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[1ZZ]↺↻

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[08V]↺↻

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[200]↺↻

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[202]↺↻

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[20T]↺↻

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[205]↺↻