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:

(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}

• 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