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Is a vector space over a ring or over a field?

What is a vector space? I can see two different formulations, and between them there is one difference: commutativity.

DEFINITION 1 (See here)

Let $(F, +_F, \times_F)$ be a division ring. Let $(\mathcal{V}, +_\mathcal{V})$ be an abelian group. Let $(\mathcal{V}, +_\mathcal{V}, \cdot)_F$ be a unitary module over $F$. Then $(\mathcal{V}, +_\mathcal{V}, \cdot)_F$ is a vector space over $F$. That is, a vector space is a unitary module over a ring, whose ring is a division ring.

DEFINITION 2

Let $(F, +_F, \times_F)$ be a field. Let $(\mathcal{V}, +_\mathcal{V})$ be an abelian group. Let $\cdot: F\times \mathcal{V} \longrightarrow \mathcal{V}$ be a function. A vector space is $(\mathcal{V}, +_\mathcal{V}, \cdot)_F$ such that $\forall a,b, \in F$ and $\forall x,y \in \mathcal{V}$:

  • $\cdot$ right distributive: $(a +_F b) \cdot x = (a\cdot x) +_\mathcal{V} (b\cdot x)$
  • $\cdot$ left distributive: $\,\,\, a \cdot (x +_\mathcal{V} y) = (a\cdot x) +_\mathcal{V} (a\cdot y)$
  • $\cdot$ compatible with $\times_F$: $(a\times_F b) \cdot x = a \cdot (b\cdot x)$
  • $\times_F$ 's identity is $\cdot$'s identity: $1_F \cdot x = x$

There could also be other definitions,but for now it doesn't matter. What matter is that commutativity is not considered in the same way in both definitions! In the first definition, we ahve a division ring (not a commutative division ring, i.e. a field!), while in the second we have a field (i.e. a commutative division ring). Just for clarity, I will write down some definitions that could be useful.


Useful Definitions

Binary Operation

A binary operation is a function $*:S\times T \longrightarrow \mathbb{U}$ from the Cartesian product of two sets, to a universe set $\mathbb{U}$. When $S=T$ we say that $*$ is a binary operation on S.

Closed Binary Operation on a Set

A closed binary operation on a set $S$ is a binary operation where $*:S\times T \longrightarrow \mathbb{U}$ where $S=T=\mathbb{U}$. That is, a closed binary operation is a function $f:S\times S \longrightarrow S$ We usually denote by $f((s_1,s_2)) = s_1 * s_2$, the result of the binary operation, for $s_1, s_2\in S$. From now on, unless explicitly mentioned (see Algebraic Structure) by "binary operation" we will mean a closed binary operation on a set.

Algebraic Structure

An algebraic structure is an ordered tuple $(S, *_1, *_2, \cdots *_n)$ where $S$ is a set which has one or more binary operations $*_1, *_2, \cdots, *_n$ defined on all elements of $S\times S$. (Note that in this case we call binary operation a function that starts from $S\times S$ but not necessarily ends in $S$, rather ends up on a universe set $\mathbb{U} \supset S$)

Magma

A magma is a set $S$ with a binary operation $*$. Notice that by definition of the (closed) binary operation, a magma is closed under its binary operation.

Semigroup

A semigroup is a set $S$ with a binary operation $*$ that is associative, i.e. $\forall a,b,c\in S \,\, a * (b * c) = (a* b) * c$ A semigroup is an associative magma.

Monoid

A monoid is a set $S$ with a binary operation $*$ such that

  • Associativity: $\forall a,b,c\in S \,\, a * (b * c) = (a* b) * c$
  • Identity: $\exists e\in S \,\, \forall a\in S: \,\, e * a = a * e = a$

A monoid is a semigroup with an identity element. A monoid is an associative magma with an identity element.

Group

A group is a set $G$ together with a binary operation $*: G\times G \longrightarrow G, \, (a, b) \longmapsto a * b$ such that the following axioms are satisfied

  • Associativity: $\forall a,b,c \in G \,\,\,\, a * (b * c) = (a * b) * c$
  • Existence of Identity / Neural Element: $\exists e \in G, \,\forall a\in G: \,\,\,\, a * e = e * a = a$
  • Existence of Inverse: $\forall a \in G, \exists b \in G : \,\,\,\, a * b = e$ where $e$ is the neural element / identity of $G$.

Abelian Group

A group $G$ is called abelian (or commutative) if $\forall a,b \in G \,\,\,\, a * b = b * a$

Ring

A ring is a set $R$ with two binary operations $$+: F\times F \longrightarrow F \qquad \text{and} \qquad\cdot :F \times F \longrightarrow F$$ often called addition and multiplication respectively, such that:

  • $(R, +)$ is an abelian group. [assoc, id, inv, commut]
  • $(R, \cdot)$ is a semigroup. [assoc]
  • $\cdot$ is distributive with respect to $+$ (both left and right distributive). This means that $\forall a,b,c \in R$ $a \cdot (b + c) = a \cdot b + a \cdot c \qquad \text{and} \qquad (b + c) \cdot a = b \cdot a + c \cdot a$

Unity Ring

A ring $(R, +, \cdot)$ where $(R, \cdot)$ is a monoid on top of being a semigroup. That is $(R, \cdot)$ has an identity element.

Division Ring

A division ring is a unity ring $(R, +, \cdot)$ where $(R, \cdot)$ is a group (not necessarily abelian!) on top of being a monoid. That is, $(R, \cdot)$ has an inverse for every element: $$\forall x\in R^\times \, : \, \exists ! x^{-1}\in R^\times \, : \, x^{-1} * x = x * x^{-1} = 1_R$$ where $R^\times := R \backslash \{0_R\}$ and $1_R$ and $0_R$ are $+$ and $\cdot$ identity elements respectively.

Field

A field is a set $F$ with two binary operations $$+: F\times F \longrightarrow F \qquad \text{and} \qquad\cdot :F \times F \longrightarrow F$$ called addition and multiplication respectively, such that:

  • $(F, +)$ and $(F\backslash\{0\}, \cdot)$ are abelian groups
  • Distributivity holds: $\forall a, b, c \in F \,\,\,\,\, a\cdot(b+c) = a\cdot b + a\cdot c \in F$ A field is just a commutative division ring.

Homomorphism

A homomorphism is a function between two algebraic structures of the same type the preserves the operations of the structures. Therefore, it is a function $f: A\longrightarrow B$ between two sets $A$ and $B$ equipped with the same structure such that, if $*$ is an operation of the structure (for instance, a binary operation) then $f(x*y) = f(x)*f(y)$ $\forall x,y \in A$

Relevant examples of homomorphisms are:

  • Semigroup homomorphism: function between semigroups preserving the semigroup operation.
  • Monoid homomorphism: function between monoids preserving the monoid operation, mapping identity element of first monoid to identity element of second monoid.
  • Group homomorphism: function between groups that preserves group operation, maps identity element of first group to identity element of second group, maps inverse of an element of first group to inverse of the image of this element. Note: semigroup homomorphism between groups is necessarily a group homomorphism. (This is because we have $f(a*e)=f(a)*f(e)$ due to semigroup homomorphism, but also $f(a*e)=f(a)$ since $e$ is identity for the group. Thus $f(a)=f(a)*f(e)$ so $e$ is mapped to $f(e)$ and identity is preserved. Similar for inverse)
  • Ring homomorphism: function between rings that preserves ring addition, ring multiplication and multiplicative identity.

Endomorphism

An endomorphism is a homomorphism where the domain and the codomain are the same.

Ring Homomorphism

A ring homomorphism is a function $f$ between two rings $R$ and $S$, i.e. $f: R\longrightarrow S$, such that $f$ is:

  • Addition preserving: $f(a+b) = f(a)+f(b)$ $\forall a,b \in R$
  • Multiplication preserving: $f(ab) = f(a)f(b)$ $\forall a,b \in R$
  • Unity preserving: $f(1_R) = 1_S$

Endomorphism Ring

Let $(G, *)$ be an abelian group. Let $\mathbb{G}$ be the set of all group endomorphisms of $(G, *)$. That is, $\mathbb{G}$ is the set of all functions $f:(G, *) \longrightarrow (G, *)$ preserving $*$, identity element and inverse elements. Now we define a new binary operation (can be seen as the composition of functions) $$\oplus : \mathbb{G}\times \mathbb{G}\longrightarrow \mathbb{G}, \,\, f\oplus g \longmapsto f \circ g$$ Then $(\mathbb{G}, *, \oplus)$ is a unity ring called ring of endomorphisms of the abelian group $(G, *)$.

Algebraic Structure over a Ring

Let $(R, +_R, \times_R)$ be a ring. Let $(S, *_1, *_2, \cdots, *_n)$ be an algebraic structure with $n$ operations. Let $\circ: R\times S \longrightarrow S$ be a binary operation. Then $(S, *_1, *_2, \cdots, *_n, \circ)_R$ is an R-algebraic structure with $n$ operations.

Unitary Module

Let $(R, +_R, \times_R)$ be a unit ring with identity $1_R$. Let $(G, +_G)$ be an abelian group (which is an algebraic structure). A \textbf{unitary module over $R$} is an algebraic structure over a ring $R$ with one operation, i.e. $(G, +_G, \circ)_R$, satisfying $\forall x,y \in G$ and $\forall \lambda, \mu \in R$

  • $\circ$ left distributive: $\lambda \circ (x +_G y) = (\lambda \circ x) +_G (\lambda \circ y)$
  • $\circ$ right distributive: $\lambda +_R \mu) \circ x = (\lambda \circ x) +_G (\mu \circ x)$
  • $\circ$ compatible with $\times_R$: $(\lambda \times_R \mu) \circ x = \lambda \circ (\mu \circ x)$
  • $\times_R$ identity element is $\circ$ identity element: $1_R \circ x = x$

Vector Space

A vector space $\mathcal{V}$ over a field $(F, +_F, \times_F)$ is an abelian group $(\mathcal{V}, +)$ together with a function $\cdot : F\times \mathcal{V} \longrightarrow \mathcal{V}$ such that $\forall a, b \in F$ and $\forall x,y \in \mathcal{V}$

  • Right Distributive: $ \,\, (a + b) \cdot x = a\cdot x + b \cdot x \in \mathcal{V}$
  • Left Distributive: $ \,\, a\cdot (x + y) = a\cdot x + a \cdot y \in \mathcal{V}$
  • Compatibility of $\cdot$ with $\times_F$ $\,\, (ab)\cdot x = a \cdot (b\cdot x)$
  • $\times_F$ identity is $\cdot$ identity: $\,\, 1\cdot x = x \in \mathcal{V}$

Alternatively, a Vector Space can be defined as follows:

Let $(F, +_F, \times_F)$ be a division ring. Let $(\mathcal{V}, +_\mathcal{V})$ be an abelian group. Let $(G, +_G, \cdot)_F$ be a unitary module over $F$. Then $(G, +_G, \cdot)_F$ is a vector space over $F$. That is, a vector space is a unitary module whose scalar ring is a division ring. If $\times_F$ is commutative then $(F, +_F, \times_F)$ is a commutative division ring, i.e. a field.

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