Phy622 Mathematical Methods 3

01-05

Algebraic Structures

• Arithmetic
• Symmetry of Groups
• Groups
• Rings

Groups

• Definition of a [Group]
• Abelian group

Rings

• Definition of a [Ring]

Examples:

• Rational Numbers
• Reals
• Integers
• $S = ({0}, *, +)$
• $S = ({a+b\sqrt 2\vert\ a, b \in \mathbb{Q}})$

• $S = ({N\times N matrices})$

• [Ring#Special Rings]

Exercises

1. Show $x\cdot 0 = 0\cdot x = 0$ on a [Ring]
2. Show $x(-y) = (-x)y = -xy$ on a [Ring]