Algebraic Structure

An algebraic structure is a set equipped with algebraic operations (addition, multiplication etc.)

Ring

A ring has addition and multiplication defined, where:

• $0$ is additive identity: $a + 0 = a$
• $1$ is multiplicative identity: $a \cdot 1 = a$
• $-a$ is additive inverse: $a + (-a) = 0$
• $+$ is associative: $(a + b) + c = a + (b + c)$
• $+$ is commutative: $a+b = b+a$
• $\cdot$ is associative: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
• $\cdot$ is distributive: $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$

Commutative Ring

A commutative ring is a ring where:

• $\cdot$ is commutative. ($a \cdot b = b \cdot a$)

Field

A field is a commutative ring where:

• $a^{-1}, \, a \ne 0$ is multiplicative inverse: $a \cdot (a^{-1}) = 1$ Field is the most intuitive one because the algebra as we learn in school works on fields. Real numbers, complex number and rational numbers are fields.