Basic Theorems Regarding Rings

# Basic Theorems Regarding Rings

Recall from the Rings page that a ring is a set $R$ with two binary operations $+$ and $*$ such that:

• $R$ is closed under $+$, $+$ is associative, $R$ contains an identity element $0$ under $+$, $R$ contains inverse elements for each element in $R$ under $+$, and $+$ is commutative.
• $R$ is closed under $*$, $*$ is associative, and $R$ contains an identity element $1$ under $*$.
• $*$ distributes over $+$.

When first looking at groups we looked at some basic theorems on the Basic Theorems Regarding Groups page. Since every ring is a group with respect to the operation $+$, we note that the theorems on that page also hold for rings. We will now look at some more basic theorems that can be proven from the ring axioms.

 Theorem 1: Let $(R, +, *)$ be a ring and let $1$ be the identity of $*$. The the identity $1$ is unique.
• Proof: Suppose that $1$ and $i$ are both identities on elements in $R$ of $*$. Then:
(1)
\begin{align} \quad 1 = 1 * i = i \end{align}
• The first equality comes from the fact that $e$ is an identity of $*$, while the second equality comes from the fact that $1$ is an identity of $*$. Therefore $1 = i$ and so the identity $1$ of $*$ is unique. $\blacksquare$
 Theorem 2: Let $(R, +, *)$ be a ring with $0$ be the identity of $+$, and $1$ be the identity of $*$. Then for all $a \in R$ we have that $a * 0 = 0$ and $0 * a = 0$.
• Proof: Let $a \in R$. Then:
(2)
\begin{align} \quad a * 0 = a * (1 + (-1)) = (a * 1) + a * (-1) = a + (-a) = 0 \end{align}
• Similarly:
(3)
\begin{align} \quad 0 * a = (1 + (-1)) * a = (1 * a) + ((-1) * a) = a + (-a) = 0 \quad \blacksquare \end{align}
 Theorem 3: Let $(R, +, *)$ be a ring with $1$ be the identity of $*$. Then $a * (-b) = (-a) * b = -(a * b)$.
• Proof: Let $a, b \in R$. Then:
(4)
\begin{align} \quad a * (-b) = a * ((-1) * b) = (a * (-1)) * b = (-a) * b = (-1) * (a * b) = -(a * b) \quad \blacksquare \end{align}