Rings Review

# Rings Review

We will now review some of the recent material regarding rings.

• On the Rings page we defined a ring to be a triple $(R, +, *)$ where $R$ is a set and $+$ and $*$ are binary operations which satisfy:
• (1) For all $a, b \in R$ we have that $(a + b \in R)$ (Closure under $+$).
• (2) For all $a, b, c \in R$, $a + (b + c) = (a + b) + c$ (Associativity of elements in $R$ under $+$).
• (3) There exists an $0 \in R$ such that for all $a \in R$ we have that $a + 0 = a$ and $0 + a = a$ (The existence of an identity element $0$ of $R$ under $+$).
• (4) For all $a \in R$ there exists a $-a \in R$ such that $a + (-a) = 0$ and $(-a) + a = 0$ (The existence of inverses for each element in $R$ under $+$).
• (5) For all $a, b \in R$ we have that $a + b = b + a$ (Commutativity of elements in $R$ under $+$).
• (6) For all $a, b \in R$ we have that $a * b = b * a$ (Closure under $*$).
• (7) For all $a, b, c \in R$, $a * (b * c) = (a * b) * c$ (Associativity of elements in $R$ under $*$).
• (8) There exists a $1 \in R$ such that for all $a \in R$ we have that $a * 1 = a$ and $1 * a = a$ (The existence of an identity element $1$ of $R$ under $*$).
• (9) For all $a, b, c \in R$ we have that $a * (b + c) = (a * b) + (b * c)$ and $(a + b) * c = (a * c) + (b * c)$ (Distributivity of $*$ over $+$).
Theorem
(a) If $(R, +, *)$ is a ring and $1$ is an identity for $*$ then this identity is unique.
(b) If $(R, +, *)$ is a ring and $0$ is the identity for $+$ and $1$ is an identity for $*$ then for all $a \in \mathbb{R}$, $a * 0 = 0$ and $0 * a = 0$.
(c) If $(R, +, *)$ is a ring and $1$ is an identity for $*$ then $a * (-b) = -(a * b)$ and $(-a) * b = - (a * b)$.
• On the Subrings and Ring Extensions page we said that if $(R, +, *)$ is a ring and if $S \subseteq R$ is such that $(S, +, *)$ is also a ring, then $(S, +, *)$ is a Subring of $(R, +, *)$ and $(R, +, *)$ is called a Ring Extension of $(S, +, *)$.
• We then look at an important criterion for determining whether a subset of a ring is a subring. We proved that if $(R, +, *)$ is a ring and $S \subseteq R$ is such that $S$ is closed under $+$, for every $a \in S$ we have that $-a \in S$, $S$ is closed under $*$, and $1 \in S$, then $(S, +, *)$ is a subring of $(R, +, *)$.
• On The Direct Product of Two Rings page we defined the direct product of two rings. We said that if $(R, +_1, *_1)$ and $(S, +_2, *_2)$ are rings, then the Direct Product of these rings is the set $T = R \times S$ with the binary operations $+$ and $*$ defined for all $(a, b), (c, d) \in T$ by:
(1)
\begin{align} \quad (a, b) + (c, d) = (a +_1 c, b +_2 d) \end{align}
(2)
\begin{align} \quad (a, b) * (c, d) = (a *_1 c, b *_2 d) \end{align}