# Ring

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A ring is an algebraic structure comprised of a set paired with two operations on the set, which are designated as addition ($+$) and multiplication ($\cdot$). As a group can be conceptualized as an ordered pair of a set and an operation, $\left(G,\cdot\right)$, a ring can be conceptualized as an ordered triple $\left(R,+,\cdot\right)$.

A set with addition and multiplication, $\left(R,+,\cdot\right)$, is a field if and only if it satisfies the following properties:

1. Commutativity of addition — For all $a, b \in R$, $a + b = b + a$;
2. Associativity of both addition and multiplication — For all $a, b, c \in R$, $\left(a + b\right) + c = a + \left(b + c\right)$ and $\left(a \cdot b\right) \cdot c = a \cdot \left(b \cdot c\right)$;
3. Additive IdentityThere exists a "zero" element, $0 \in R$, called an additive identity, such that $a + 0 = a = 0 + a$, for all $a \in R$;
4. Additive Inverses — For each $a \in R$, there exists a $b \in R$, called an additive inverse of $a$, such that $a + b = 0$;
5. Distributive property — For all $a, b, c \in R$, $a \cdot \left(b + c\right) = a \cdot b + a \cdot c$ and $\left(b + c\right) \cdot a = b \cdot a + c \cdot a$;
6. Closure of addition and multiplication — For all $a, b \in R$, $a + b \in R$ and $a \cdot b \in R$.

A prototype of these kind of structures in the ring of the integers, $(\mathbb{Z},+,\cdot)$.

We will often abbreviate the multiplication of two elements, $a \cdot b$, by juxtaposition of the elements, $ab$. Also, when combining multiplication and addition in an expression, multiplication takes precedence over addition unless the addition is enclosed in parenthesis. That is, $a + bc + d = a + \left(bc\right) + d$.

We can also denote $-a$ as the additive inverse of any $a \in R$. Furthermore, we can define an operation, called subtraction by $a - b = a + \left(-b\right)$.

## Optional Properties

Optionally, a ring $R$ may have additional properties:

• We define $R$ to be a commutative ring if the multiplication is commutative: $a \cdot b = b \cdot a$ for all $a,b\in R$;
• We define $R$ to be a ring with unity if there exists a multiplicative identity $1\in R$: $1\cdot a = a = a \cdot 1$ for all $a \in R$;
• Furthermore, a commutative ring with unity $R$ is a field if every element except 0 has a multiplicative inverse: For each non-zero $a \in R$, there exists a $b\in R$ such that $a\cdot b = b \cdot a = 1$;
• We define $R$ to be an integral domain if it is commutative, has unity, and the zero product rule holds: $a \cdot b = 0$ implies either $a = 0$ or $b = 0$ for all $a,b \in R$.

## Important Results

From the given axioms for a ring $R$, it can be deduced that:

• $\left(R,+\right)$ is an abelian group;
• $0 \cdot a = 0$, for all $a \in R$;
• $a\left(-b\right) = \left(-a\right)b = -\left(ab\right)$, for all $a, b \in R$;
• $\left(-a\right)\left(-b\right) = ab$, for all $a, b \in R$;
• If $R$ has unity $1$:
• $\left(-1\right) \cdot a = -a$, for all $a \in R$;
• $\left(-1\right) \cdot \left(-1\right) = 1$;
• Multiplication distributes over subtraction.