# Group, Ring, Field, Module, Vector Space

This page is a sub-page of our page on Some basic algebraic concepts.

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Related pages:

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A binary composition $\ast$ on a set $\mathcal S$ is a function

${\mathcal S \times \mathcal S \ni (x, y) \, \mapsto \, x \ast y \in \mathcal S.}$

The composition $\ast$ is called associative if

$(x \ast y) \ast z = x \ast (y \ast z) \, , \forall x, y, z \in \mathcal S$

and $\ast$ is called commutative if

$x \ast y = y \ast x \, , \forall x, y \in \mathcal S.$

If there exists an element $e \in \mathcal S$ such that

$e \ast x = x \ast e = x \, , \forall x \in \mathcal S$

then $e$ is called a unit (or unit element).

If $e$ and $e'$ are unit elements, then we have

$e \ast e' = e = e'$

Hence there is at most one unit element, i.e., a unit element is unique (if it exists).

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Assume that $(\mathcal S, \ast)$ has a unit element $e.$ Then the element $x \in \mathcal S$ is said to have:

a left inverse, if there is an element $y \in \mathcal S$ such that $y \ast x = e.$

a right inverse, if there is an element $y \in \mathcal S$ such that $x \ast y = e.$

an inverse , if there is an element $y \in \mathcal S$ such that $x \ast y = y \ast x = e.$

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Let $\ast$ and $\bullet$ be two binary compositions on $\mathcal S.$ We say that:

$\ast$ is left-distributive over $\bullet$ if

$(x \ast ( y \bullet z) = (x \ast y) \bullet (x \ast z) \, , \forall x, y, z \in \mathcal S.$

$\ast$ is right-distributive over $\bullet$ if

$(y \bullet z) \ast x = (y \ast x) \bullet (z \ast x) \, , \forall x, y, z \in \mathcal S.$

$\ast$ is distributive over $\bullet$ if $\ast$ is both left- and right-distributive over $\bullet.$

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Using the concepts defined above,
we can define a number of important algebraic structures:

• Semigroup: A set $\mathcal S$ with a binary composition $\ast$ that is associative.

• Monoid: A semigroup which has a unit element.

• Group: A monoid where each element has an inverse.

• Abelian group: A group whose binary composition is commutative.

• Ring: A set $\mathcal R$ with two binary compositions,
called addition (denoted by $+$)
and multiplication (denoted by $\cdot$ or by juxtaposition)
such that $(\mathcal R, +)$ is an abelian group,
$(\mathcal R, \cdot)$ is a semigroup,
and multiplication is distributive over addition.

• Ring with unit: A ring $\mathcal R$ where $(\mathcal R, \cdot)$ is a monoid.

• Commutative ring:  A ring with commutative multiplication.

• Field: A commutative ring with unit
where each non-zero element has a multiplicative inverse.

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Semantics of the concepts Group, Ring, and Field:

Semantics of the concepts Group, Ring, and Field (Semigroup):

Semantics of the concepts Group, Ring, and Field (Monoid):

Semantics of the concepts Group, Ring, and Field (Group):

Semantics of the concepts Group, Ring, and Field (Abelian group):

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Module: An abelian group $(\mathcal M, \oplus)$ is called a module over the ring $(\mathcal R, +, \cdot),$ or an $\mathcal R$ module, if there is a function

${\mathcal R \times \mathcal M \ni (r, m) \, \mapsto \, r m \in \mathcal M}, \text{ such that}$

(i):   $0m = 0$ where $0$ is the additive unit of  $\mathcal R.$

(ii): $1m = m$ if $\mathcal R$ has a multiplicative unit $1.$

(iii): $(r + r') m = (r m) \oplus (r' m).$

(iv): $r ( m \oplus m') = (r m) \oplus (r m').$

(v): $(r \cdot r') m = r (r' m).$

Note: Most often the additive unit of $\mathcal R$ and the additive unit of $\mathcal M$ are both denoted by $0.$ Moreover, the direct sum $\oplus$ is denoted by $+$ and is referred to as addition.

Vector space: A module over a field.

Group homomorphism: A function ${\varphi : \mathcal G \, \rightarrow \, \mathcal G'}$ where $(\mathcal G, \ast)$ and $(\mathcal G', {\ast}')$ are groups, and

$\varphi ( x \ast y) = \varphi (x) {\ast}' \varphi (y), \, \forall x, y \in \mathcal G.$

Module homomorphism: A function ${\varphi : \mathcal M \, \rightarrow \, \mathcal M'}$ where $\mathcal M$ and $\mathcal M'$ are $\mathcal R$-modules and

$\varphi ( rm_1 + m_2) = r \varphi (m_1) + \varphi (m_2), \, \forall r\in \mathcal R, \, \forall m_1, m_2 \in \mathcal M.$

Note: The $+$ sign to the left represents the addition in $\mathcal M,$ while the $+$ sign to the right represents the addition in $\mathcal M'.$ Strictly speaking, the latter addition should be written as $+'.$ This is a typical example of the kind of relaxation of notation that is often used in mathematics. The different meanings of a given symbol is most often clear from the context. If not, the more precise (but more cumbersome) notation can always be brought in to clear up the confusion.

## 3 thoughts on “Group, Ring, Field, Module, Vector Space”

You are quite right. The mathrehab site is very much “work in progress” and this page will be a reference page with “crisp definitions” of (some) common mathematical constructs. They will be referenced from other – hopefully more explanatory – pages on this site.

Best regards
/Ambjörn

P.S: Also, we recently updated the “template” for this site, and some of the LaTEX formulas don’t seem to work (yet) in the new setting.

2. Lokesh mogili says:

Hi thank you .super

3. Mull says:

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