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 $+$).

- We then look at some basic results regarding rings on the
**Basic Theorems Regarding Rings**page. We summarize these results below:

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:

\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}

- We verified that $(T, +, *)$ is indeed a ring, and then looked at an example on
**The Direct Product of the Ring of m x m and n x n Matrices**page.

- We lastly looked at some examples of rings and subrings:
**The Ring of Real and Complex Numbers****The Ring of Polynomials with Real Coefficients****The Ring of Polynomials with Ring Coefficients****The Ring of n x n Matrices****The Ring of Gaussian Integers Z(i)****The Ring of Q(√2)****The Ring of Z/2Z****The Ring of Z/nZ****The Subring of n x n Upper and Lower Triangular Matrices****The Subring of Polynomials with Subring Coefficients**