Groups, Rings, Fields Cheatsheet

Cheatsheet Content

1. Definitions of Number Sets Natural Numbers ($\mathbb{N}$): The counting numbers. $\mathbb{N} = \{1, 2, 3, \ldots\}$. (Some definitions include 0: $\mathbb{N}_0 = \{0, 1, 2, 3, \ldots\}$). Whole Numbers ($\mathbb{W}$): Natural numbers including zero. $\mathbb{W} = \{0, 1, 2, 3, \ldots\}$. Integers ($\mathbb{Z}$): All whole numbers and their negatives. $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$. Rational Numbers ($\mathbb{Q}$): Numbers that can be expressed as a fraction of two integers. $\mathbb{Q} = \{a/b \mid a, b \in \mathbb{Z}, b \neq 0\}$. Irrational Numbers: Numbers that cannot be written as a simple fraction (e.g., $\sqrt{2}$, $\pi$, $e$). Real Numbers ($\mathbb{R}$): All numbers on the number line, including rational and irrational numbers. $\mathbb{R} = \mathbb{Q} \cup \text{(Irrational Numbers)}$. Complex Numbers ($\mathbb{C}$): Numbers with a real part and an imaginary part. $\mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}, i^2 = -1\}$. 2. Algebraic Structures Overview This table summarizes common number sets and their properties under addition (+) and multiplication ($\times$) to form algebraic structures like Groups, Rings, and Fields. Number Set Group under + Group under $\times$ (nonzero) Ring? Field? $\mathbb{N}$ (Natural Numbers) ❌ ❌ ❌ ❌ $\mathbb{Z}$ (Integers) ✔️ ❌ ✔️ ❌ $\mathbb{Q}$ (Rational Numbers) ✔️ ✔️ ✔️ ✔️ $\mathbb{R}$ (Real Numbers) ✔️ ✔️ ✔️ ✔️ $\mathbb{C}$ (Complex Numbers) ✔️ ✔️ ✔️ ✔️ 3. Definitions of Algebraic Structures Group A set $G$ with a binary operation $*$ is a group $(G, *)$ if it satisfies the following axioms: Closure: For all $a, b \in G$, $a * b \in G$. Associativity: For all $a, b, c \in G$, $(a * b) * c = a * (b * c)$. Identity Element: There exists an element $e \in G$ such that for all $a \in G$, $e * a = a * e = a$. Inverse Element: For each $a \in G$, there exists an element $a^{-1} \in G$ such that $a * a^{-1} = a^{-1} * a = e$. If additionally $a * b = b * a$ for all $a, b \in G$, it is an Abelian Group (Commutative Group). Ring A set $R$ with two binary operations, addition $(+)$ and multiplication $(\times)$, is a ring $(R, +, \times)$ if it satisfies the following axioms: $(R, +)$ is an Abelian Group. Closure under multiplication: For all $a, b \in R$, $a \times b \in R$. Associativity of multiplication: For all $a, b, c \in R$, $(a \times b) \times c = a \times (b \times c)$. Distributivity: For all $a, b, c \in R$: $a \times (b + c) = (a \times b) + (a \times c)$ (left distributivity) $(b + c) \times a = (b \times a) + (c \times a)$ (right distributivity) A ring is a Commutative Ring if multiplication is commutative ($a \times b = b \times a$). A ring has an Identity Element (Unity) if there exists $1 \in R$ such that $1 \times a = a \times 1 = a$ for all $a \in R$. Field A set $F$ with two binary operations, addition $(+)$ and multiplication $(\times)$, is a field $(F, +, \times)$ if it satisfies the following axioms: $(F, +)$ is an Abelian Group. $(F \setminus \{0\}, \times)$ is an Abelian Group (where $0$ is the additive identity). Distributivity: Multiplication distributes over addition. In simpler terms, a field is a commutative ring with unity where every non-zero element has a multiplicative inverse. 4. Number Sets as Algebraic Structures Groups Formed by Number Sets $(\mathbb{Z}, +)$, $(\mathbb{Q}, +)$, $(\mathbb{R}, +)$, $(\mathbb{C}, +)$ are all Abelian Groups under addition. The sets $\mathbb{Q} \setminus \{0\}$, $\mathbb{R} \setminus \{0\}$, $\mathbb{C} \setminus \{0\}$ are Abelian Groups under multiplication. $\mathbb{N}$ and $\mathbb{N}_0$ are NOT groups under addition because they lack additive inverses (e.g., $3 \in \mathbb{N}$ has no $-3 \in \mathbb{N}$). $\mathbb{Z}$ is NOT a group under multiplication because most elements lack multiplicative inverses (e.g., $2 \in \mathbb{Z}$ has no $1/2 \in \mathbb{Z}$). Rings Formed by Number Sets $\mathbb{Z}$ (Integers) forms a Commutative Ring with Unity . It is a standard example of a ring. $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ are all Rings . More specifically, they are Fields because every non-zero element has a multiplicative inverse. $\mathbb{N}$ and $\mathbb{N}_0$ are NOT rings because addition does not form a group (lacks additive inverses).