FoldUnfold Table of Contents Ring Automorphisms Ring Automorphisms We have already defined Group Automorphisms. We now extend this concept to rings. Definition: Let $(R, +, *)$ be a ring with multiplicative identity $1$. A function $\phi : R \to R$ is a Ring Automorphism if: 1) $\phi (a + b) = \phi (a) + \phi […]

## Ring Isomorphisms

FoldUnfold Table of Contents Ring Isomorphisms Ring Isomorphisms We have already defined Group Isomorphisms. We now extend the concept to rings. Definition: Let $(R, +_1, *_1)$ and $(S, +_2, *_2)$ be rings with multiplicative identities $1_R$ and $1_S$ respectively. A function $\phi : R \to S$ is a Ring Isomorphism if: 1) $\phi(a +_1 b) […]

## Ring Homomorphisms

FoldUnfold Table of Contents Ring Homomorphisms Ring Homomorphisms We have already defined Group Homomorphisms. We now extend the concept to rings. Definition: Let $(R, +_1, *_1)$ and $(S, +_2, *_2)$ be rings with multiplicative identities $1_R$ and $1_S$ respectively. A function $\phi : R \to S$ is a Ring Homomorphism if: 1) $\phi(a +_1 b) […]

## Quotient Rings

FoldUnfold Table of Contents Quotient Rings Quotient Rings Recall from the Ideals of Rings page that if $(R, +, *)$ is a ring then $I$ is said to be an ideal of $(R, +, *)$ if: 1) $(I, +)$ is a subgroup of $(R, +)$. 2) For all $r \in R$ and for all $i […]

## Ideals of Rings

FoldUnfold Table of Contents Ideals of Rings Ideals of Rings Definition: Let $(R, +, *)$ be a ring. An Ideal (or Two-sided Ideal) is a subset $I \subseteq R$ such that: 1) $(I, +)$ is a subgroup of $(R, +)$. 2) For all $r \in R$ and for all $i \in I$ we have that […]

## The Integral Domain of Z/pZ

FoldUnfold Table of Contents The Integral Domain of Z/pZ The Integral Domain of Z/pZ Recall from the Integral Domains page that a ring $(R, +, *)$ is said to be an integral domain if it is a commutative ring and $R$ contains no zero divisors, that is, if $0$ is the identity of $+$ then […]

## The Cancellation Property of * on Integral Domains

FoldUnfold Table of Contents The Cancellation Property of * on Integral Domains The Cancellation Property of * on Integral Domains Recall from the Integral Domains page that an integral domain is a commutative ring $(R, +, *)$ that contains no zero divisors. That is, the ring $(R, +, *)$ satisfies all of the ring axioms […]

## Integral Domains

FoldUnfold Table of Contents Integral Domains Integral Domains Recall from the Zero Divisors in Rings page that if we consider a ring $(R, +, *)$ where $0$ is the identity of $+$ then a zero divisor of $R$ is an element $a \in R \setminus \{ 0 \}$ such that there exists an element $b […]

## The Ring of Z/nZ

FoldUnfold Table of Contents The Ring of Z/nZ The Ring of Z/nZ Recall from the Rings page that if $+$ and $*$ are binary operations on the set $R$, then $R$ is called a ring under $+$ and $*$ denoted $(R, +, *)$ when the following are satisfied: 1. For all $a, b \in R$ […]

## The Ring of Z/2Z

FoldUnfold Table of Contents The Ring of Z/2Z The Ring of Z/2Z Recall from the Rings page that if $+$ and $*$ are binary operations on the set $R$, then $R$ is called a ring under $+$ and $*$ denoted $(R, +, *)$ when the following are satisfied: 1. For all $a, b \in R$ […]

- « Previous Page
- 1
- 2
- 3
- 4
- 5
- …
- 25
- Next Page »