MATH 415 Lecture 15

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Rings

Ring Homomorphisms

Recall that $\phi$ is a homomorphism of a ring $R$ if and only if $\phi :R\to R'$ satisfies the homomorphism property over both addition and multiplication.

For example, $\phi :\mathbb {Z} \to \mathbb {Z} _{n}$ where $\phi (m)=m{\pmod {n}}$ is a homomorphism. $\phi$ is obviously a homomorphism over addition, but multiplication can be trickier to see.

$\phi (x\cdot y)=x\cdot y{\pmod {n}}$. It is a property of modular arithmetic that the mod of a product is equivalent to the product of the mods, so $x\cdot y{\pmod {n}}=\left(x\mod {n}\right)\cdot \left(y\mod {n}\right)=\phi {x}\cdot \phi {y}$.

Ring Isomorphisms

Recall the definition of an isomorphism:

$\phi :R\to R'$ is an isomorphism if it satisfies all of the following properties:

1. $\phi$ is a homomorphism
2. $\phi$ is one-to-one (injective)
3. $\phi$ is onto (surjective)

Or, more simply stated, $\phi$ is a bijective homomorphism.

Isomorphisms define an equivalence relation:

1. Reflexive. $R\simeq R$ via $\phi (x)=x$ (identity function)
2. Symmetric. $R\simeq R'$ via $\phi :R\to R'$. Then $\phi ^{-1}:R'\to R$ is also an isomorphism, hence $R'\simeq R$.
3. Transitive. $R\simeq R'$ and $R'\simeq R''$. Define composition $\zeta =\psi \circ \phi$

Warning

$\left\langle \mathbb {Z} ,+\right\rangle \simeq \left\langle 2\mathbb {Z} ,+\right\rangle$, but $\left\langle \mathbb {Z} ,+,\cdot \right\rangle \not \simeq \left\langle 2\mathbb {Z} ,+,\cdot \right\rangle$. This is because $\left\langle \mathbb {Z} ,+,\cdot \right\rangle$ has one unity, but $\left\langle 2\mathbb {Z} ,+,\cdot \right\rangle$ has no unity.

(Unity is a structural property of rings)

Unity

The unity is the multiplicative identity element in a ring

For nontrivial groups, multiplicative unity 1 is defined as such: if $x\cdot 1=1\cdot x=x$ for all $x\in R$, where $1\neq 0$ (0 is additive identity).

In the degenerate case, $\left\{0\right\}$ is called the zero ring:

• 0 acts as the additive unity ($0+0=0$), and
• the multiplicative unity ($0\cdot 0=0$)

Definitions and Observations

• $R$ is commutative if $x\cdot y=y\cdot x$ for all $x,y\in R$.
• A ring with a multiplicative identity element is a ring with unity. (denoted $1$)

$(\underbrace {1+\dots +1} _{n})(\underbrace {1+\dots +1} _{m})=(\underbrace {1+\cdots +1} _{n\cdot m})$ because $(n\cdot 1)\cdot (m\cdot 1)=(m\cdot n)\cdot 1$.

Direct Product of Rings

If $\gcd {(r,s)}=1$, then the rings $\mathbb {Z} _{r\,s}$ and $\mathbb {Z} _{r}\times \mathbb {Z} _{s}$ are isomorphic.

We already know from groups that $\mathbb {Z} _{r}\times \mathbb {Z} _{s}\simeq \mathbb {Z} _{r\,s}$ if and only if $\gcd {(r,s)}=1$.

Unities $1\in \mathbb {Z} _{r\,s}$ and $(1,1)\in \mathbb {Z} _{r}\times \mathbb {Z} _{s}$ are both generators. Therefore, $\phi (x)=(x,x)$ is an isomorphism.

In general, The unity in $R_{1}\times \dots \times R_{n}$ is constructed as $\left(1_{R_{1}},\ldots ,1_{R_{n}}\right)$, where $1_{R_{i}}\in R_{i}$

Units

$b$ is called a multiplicative inverse of $a$ if $a\,b=b\,a=1$. We denote $b=a^{-1}$.

If $R$ is a ring with unity $1\neq 0$, an element $u\in R$ is a unit of $R$ if it has a multiplicative inverse.

Note: Unity is not the same as a unit

• If $u$ is a unit, then $u^{-1}$ is a unit.
• If $u_{1}$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_2} are units, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_1 \cdot u_2} is also a unit (and thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (u_1 \cdot u_2)^{-1} = u_2^{-1} \cdot u_1^{-1}} )

Collection of units Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} forms a group that is a subgroup of the semigroup $\left\langle R,\cdot \right\rangle$.

Finding Units of a Ring

For example, find the units in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_{14}}

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} is obviously a unit,
• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} is a unit, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1 \equiv 13 \pmod{14}} is also a unit
• $(3)\cdot (5)=15\equiv 1{\pmod {14}}$, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5} are units.
• etc.

In general, units of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_n} are elements that are coprime to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .

Division Rings

If every nonzero element is a unit, then $R$ is a division ring (or a skew field)

Note that $\mathbb {Z}$ is not a division ring.

Subrings

If $R'\subseteq R$ and $R'$ is a ring, then $R'$ is a subring of $R$.

Fields

A field is a commutative division ring.

Examples: $\mathbb {Q}$, $\mathbb {R}$, $\mathbb {C}$.

Integral Domains

We start oddly with a quadratic equation $x^{2}-5x+6=(x-2)(x-3)=0$. This has solutions $x\in \left\{2,3\right\}$ in $\mathbb {R}$, but what about in $\mathbb {Z} _{12}$? $x\in \left\{2,3,6,11\right\}\subset \mathbb {Z} _{12}$.

Zero Divisors

If $a$ and $b$ are two nonzero elements of a ring $R$ such that $a\,b=0$, then $a$ and $b$ are divisors of $0$ (or $0$ divisors)

Note that $4$ and $3$ are zero divisors in $\mathbb {Z} _{12}$ since $4\cdot 3\equiv 0{\pmod {12}}$.

Theorem 19.3

In $\mathbb {Z} _{n}$, the divisors of 0 are precisely those nonzero elements that are not relatively prime to $n$.

Proof. Take $m\in \mathbb {Z} _{n}$, where $m\neq 0$. Let $d=\gcd {(m,n)}>1$, so $m$ and $n$ are both divisible by $d$. Then $m\,\left({\frac {n}{d}}\right)=\left({\frac {m}{d}}\right)\,n\equiv 0{\pmod {n}}$. Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is a zero divisor.

Now suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gcd{(m,n)} = 1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \, s = 0} for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \in \mathbb{Z}_n} . Then $m\,s=0$ implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \mid m\,s} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \mid s} since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} are relatively prime. This means that $s\equiv 0{\pmod {n}}$.

quod erat demonstrandum

Corollary. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is prime, then $\mathbb {Z} _{p}$ has no zero-divisors. (because all integers are coprime to a prime number)

Cancellation Law

Recall that (left) cancellation law holds if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \, b = a \, c} implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = c} . (similarly right)

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b,c \in R} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a ring, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\,b = a\,c} can be rewritten as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \, b - a \, c = a \, (b-c) = 0} . Thus either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b-c} is a zero-divisor, and thus the cancellation law does not hold in general (cannot divide by zero). However,

Theorem 19.5

The left and right cancellation laws hold in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} has no zero divisors.

Definition of Integral Domain

An integral domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} is a commutative ring with unity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \ne 0} containing no zero-divisors. (huh?)

Examples: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}} . In general, any field is an integral domain: (See #Theorem 19.9→)

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} \times \mathbb{Z}_p} is not an integral domain even though Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_p} are both integral domains. This is because we can choose two elements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = (1,0)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = (0,1)} in the direct product such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \cdot b = (1,0) \cdot (0,1) = (1 \cdot 0, 0 \cdot 1) = (0,0) = 0 \in \mathbb{Z} \times \mathbb{Z}_p}

Similarly, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} is an integral domain, but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_n(D)} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} matrices formed from elements of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} ) is not.

Theorem 19.9

Every field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is an integral domain

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a field, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R \setminus \left\{ 0 \right\}} is a group of units, all of which have an inverse so the cancellation laws hold.

Proof. Assume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \, b = 0} , then if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ne 0} , we can find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \ne 0} . Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \cdot (a \cdot b) = 0} is equivalent to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a^{-1} \cdot a) \cdot b = 0} , thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = 0} .