1
vote
0answers
28 views

Definition - Logarithmic Zeros and Algebraic Vector Fields

In Algebraic Geometry, what is the difference between an algebraic vector field and a derivation (are they completely different or synonymous)? What does it mean to say an algebraic vector field has ...
2
votes
1answer
71 views

Do those 'groups' have a name?

Is there a name for 'groups' that only have a neutral element on the right and an inverse for each element on the right ? If there is, does that name also hold for a neutral elt on the right and an ...
1
vote
1answer
34 views

Localization of the Integer Ring

Let $\mathbb{Z}$ be the ring of integers and let $p$ be a prime, then the $p$-localization of $\mathbb{Z}$ is defined as $\mathbb{Z}_{(p)}=\{\displaystyle\frac{a}{b}|a,b\in\mathbb{Z},p\nmid b\}$. I ...
23
votes
6answers
2k views

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...
1
vote
1answer
46 views

When is a binary operation bipotent?

I learnt that $\max(-,-)$ is a bipotent binary operation but I'm not able to find a definition of bipotent operation. QUESTION A binary operation $*:M\times M \rightarrow M$ is bipotent if ...
0
votes
1answer
37 views

On the definition of commutators

We know that given a group $G$, for every $x,y\in G$ we define $[x,y]:=x^{-1}y^{-1}xy$, the commutator of $x$ and $y$. I saw something more general, commutators involving more than two elements, like ...
1
vote
0answers
32 views

“Identify” terminology

What does it mean to "identify" two objects in algebra, as distinct from having an isomorphism or automorphism between them? I was led to think about this while reading Stewart's Galois Theory, where ...
1
vote
0answers
37 views

Binary operation (english) terminology

Foreword: I have read R.H. Bruck's A Survey of binary systems, where the notion of halfoperation is given. A halfoperation $\ast$ differs from a (binary) operation since $a\ast b$ may not be defined ...
0
votes
0answers
38 views

Difference between a vector space and an algebra

I'm new to the subject of algebras and I would like to get a better understanding of what they are exactly. Am I right to say the follwing: ...
2
votes
0answers
59 views

Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
-1
votes
2answers
56 views

A question in Isomorphism

Let G be a cyclic group. Soppose G and G' are isomorphic groups. Show that G' is also cyclic. Can Someone Solve this pleaase? I have an exam 2 hours later!
0
votes
1answer
22 views

what is the definition of linearly independent subset of abelian group?

WHat is the definition of linearly independent subset of abelian group? I know the concept of this, but i don't know how to define this term precisely. Below is what i tried to formulate: Let ...
0
votes
1answer
35 views

What is the definition of “disjoint cycles”?

I'm the one who thinks clear definition(clear with meta-language) is very important for doing mathematics. Below, i list my definitions for cycle and orbit. Let $X$ be a nonempty set. Let ...
0
votes
2answers
41 views

What is “the orbit of a permutation”? Is the term “orbit” consistent with that for group action?

reference: What is the orbit of a permutation? To be honest, i don't understand the answer in the link. The orbit of a group action is defined as follows: Let $G$ be a group acting on a set $X$. ...
0
votes
1answer
46 views

What is the meaning of an algebra?

An algebra $A(*,\hat{} ,\sim)$ is said to be Boolean algebra if it satisfies some conditions...In this statement what is the meaning of starting word an algebra?
2
votes
2answers
69 views

Tensor product is o?

$K$ generated by $(ar,rb)$ for every $a\in A$, $b\in B$, $r\in R$. $F$ generated by $r(a,b)$ for every $a\in A$, $b\in B$, $r\in R$. $(a,rb)=(ar,b)=r(a,b)$,then $K=F$, $F/K=0$ Apparently I ...
2
votes
1answer
24 views

Why do we define product of morphisms in this way

I'm always forget the definition of product of morphisms in a category, maybe the main reason is because I don't know the motivation beyond the definition: I need help to see this abstract ...
2
votes
1answer
51 views

Embedding and monomorphism

What is the difference between an embedding and a monomorphism? As far as I can see, most introductory abstract algebra texts treat them as if they are the same, i.e. an injective function from one ...
6
votes
3answers
193 views

Definition of General Associativity for binary operations

Let's say we are talking about addition defined in the real numbers. Then, by induction we define $\sum_{i=0}^{0}a_i=a_0$ and $\sum_{i=0}^{n}a_i=\sum_{i=0}^{n-1}a_i+a_n$ for $n> 1.\:$ Now, how do ...
0
votes
0answers
47 views

Definition of self-adjoint endomorphism

let $f \in End_K(E)$, and $g: (E \times E)\to \Bbb{R}^1$ a symmetric bilinear form positive definite, $f$ is self-adjoint endomorphism if $$\forall v,w \in E(g(f(v),w)=g(v,f(w)))$$ It is correct? ...
0
votes
2answers
42 views

In a commutative algebraic theory, do all constant symbols necessarily represent the same value?

Let $T$ denote a commutative algebraic theory with two nullary function symbols $a$ and $b$ (i.e. constants). Is it an automatic consequence of the definitions that $a=b$ is a theorem of $T$? My ...
1
vote
1answer
54 views

Definition of $p$-torsion module

Let $p$ be a prime. What is the definition of a $p$-torsion module ? I have googled but was not convinced.
1
vote
0answers
65 views

Understanding a.. weird definition

I came across the following definitons: Let $n$ be a positive integer and $\zeta_n$ be a primitive $n^{th}$ root of unity. For any prime $p\nmid n$, let the degree of the extension ...
3
votes
1answer
78 views

Has anyone succeeded in formalizing the notion of a complete vector space? (Not using topological ideas).

In order theory, we have the concept of a lattice, which is defined as consisting of an underlying set $L$ together with two binary operations $\wedge$ and $\vee$. Now when $L$ is finite, the concept ...
0
votes
1answer
50 views

What is the name of this theorem

Let $f: G\to K$ a morphism of groups. If $H\subset \ker f$, then there existe a unique morphism of groups $g : G/H\to K$ such that $f=gs$. Moreover, $g$ is surjective if $f$ is surjective ; $g$ ...
0
votes
2answers
72 views

Different ways of defining Absolute Value

Calculus I presents this definition of absolute value: $$f(x)=y=|x|=\left\{\begin{array}{}\;\;\;x&\text{ if}\,\,\,x\geq 0\\-x&\text{ if}\,\,\,x<0\end{array}\right.$$ But you can also ...
0
votes
1answer
89 views

About definition of “ordered semi-ring”

I need the definition of "ordered semi-ring". Can I use these properties: $a \preceq b \to a + c \preceq b + c$ $0 \preceq a \wedge 0 ≤ b \to 0 \preceq a \cdot b$ (or: $a \preceq b \wedge 0 ...
0
votes
3answers
99 views

Definition of semi-ring homomorphism

I need the definition of semi-ring homomorphism. Thanks in advance!
0
votes
1answer
313 views

Definition: finite type vs finitely generated

The mathematical term "finite type" appears more and more in the modern articles nowadays. But it is still hard to be found in the standard textbooks. I learned the definition of it from Stacks ...
0
votes
2answers
48 views

How do we know we can do cancellation in $\mathbb{Z}$?

For example, $2*x = 2*5$ implies $x = 5$ but how come, if $2$ doesn't have an inverse?
0
votes
2answers
314 views

What is an indeterminate in a polynomial ring?

I am currently studying polynomial ring. I have a basic doubt. What is $x$ in a polynomial? A polynomial is an expression $\sum_{k = 0}^n a_k x^k$, where $n,k \in \mathbb{N}$ and $a_k \in R$ where ...
5
votes
1answer
220 views

The Degree of Zero Polynomial

I wonder why the degree of the zero polynomial is $-\infty$ ? I heard that, it is $-\infty$ to make the formula $\deg(fg)=\deg(f)+\deg(g)$ hold when one of these polynomials is zero. However, if that ...
1
vote
1answer
78 views

What does it mean to be compatible with the isomorphism structure of a class?

Let $\mathrm{UR}$ denote the class of all unital rings and $\mathrm{Set}$ denote the class of all sets. Actually, perhaps it would be better to view $\mathrm{UR}$ as the groupoid whose objects are ...
1
vote
1answer
68 views

Generator of all congruence classes

Is it possible for $\langle a \rangle =\mathbb Z/n\mathbb Z^*$ for $a\in \mathbb Z/n\mathbb Z^*$? I think I recall hearing it is, what is the name this element takes? I remember hearing generator ...
2
votes
1answer
64 views

What is the function that is not a binary function called?

A binary operation is a calculation involving two elements of the set and returning another element of the set. Suppose it doesn't return an element of the set. What is the function called? For ...
4
votes
2answers
452 views

What does “characteristic” mean in mathematics?

In Germany, I have heard "ABC is 'characteristic' for XYZ" sometimes from math students. It was used like "if you know ABC, then you know you're talking about XYZ" or "ABC describes XYZ completely". ...
2
votes
1answer
71 views

Group of finite ideles

A simple question: If $\overline{\mathbb{Q}}$ denotes the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$ then what is the definition of the group of finite ideles of $\overline{\mathbb{Q}}$? ...
2
votes
2answers
213 views

What are authoritative publications regarding foundational mathematics?

I have a computer science background. In our world, there usually is an organization publishing standard documents for certain areas (e.g. W3C has Web standards, IETF publishes Internet-related ...
1
vote
2answers
63 views

“uniquely written” definition

I'm having troubles with this definition: My problem is with the uniquely part, for example the zero element: $0=0+0$, but $0=0+0+0$ or $0=0+0+0+0+0+0$. Another example, if $m \in ...
1
vote
1answer
35 views

Help with this definition of $(G:_M I)$

I didn't understand why in this definition $I$ has to be an ideal to make sense. REMARK This is from Steps in Commutative Algebra, page 107. Thanks a lot
2
votes
1answer
105 views

Understanding equivalent definitions of left cosets

I understand the standard definition of left coset, but the one I do not understand (or see why it is advantageous) is the definition that follows: Let $H\leq G$. Then a left coset of $H$ is a ...
2
votes
1answer
183 views

What's the difference between Abstract Algebra and Group Theory?

I'm slowly beginning a student of certain higher mathematics. I'm trying to see if I would prefer to study Group Theory or Abstract Algebra. I know that Abstract Algebra seems to "come before" ...
8
votes
2answers
572 views

Definition of a monoid: clarification needed

I'm only in high school, so excuse my lack of familiarity with most of these terms! A monoid is defined as "an algebraic structure with a single associative binary operation and identity element." ...
2
votes
3answers
134 views

What's the difference between $(\mathbb Z_n,+)$ and $(\mathbb Z_n,*)$?

I noticed that $(\mathbb Z_n,+)$ and $(\mathbb Z_n,*)$ are not the same thing. For example 2 is not invertible in $(\mathbb Z_6,*)$. However, since $\bar2+\bar4=\bar0$, thus it is invertible in ...
0
votes
1answer
67 views

Why Integral domains haven't an unified definition? [duplicate]

We can define integral domains as: rings without zero divisors commutative rings without zero divisors commutative rings with identity and without zero divisors I don't know why integral domains ...
0
votes
2answers
42 views

Definition of localization of rings

I'm trying to understand this definition of Hungerford's book: The definition is simple, I think I understood what the author means, but... What is $P_P$? because we will have $P_P=S^{-1}P$, ...
2
votes
1answer
140 views

What is the exact definition of polynomial functions?

I'm trying to understand the difference between polynomial functions and the evaluation homomorphisms. I noticed that I don't know what's the exact definition of a polynomial function, although I've ...
0
votes
1answer
165 views

Mapping on a set with respect to function composition

In Isaacs' Algebra, I found the following exercise Let $G$ be a group of mappings on a set $X$ with respect to function composition. Find an example where $G$ is not a subset of $\text{Sym}(X)$ and ...
2
votes
2answers
97 views

Help to understand the ring of polynomials terminology in $n$ indeterminates

In the Hungerford's book, page 150, the author defines a ring of polynomials in "n" indeterminates in the following manner: After the author defines the operations in this ring with a theorem: ...
2
votes
2answers
97 views

Why the terms “unit” and “irreducible”?

I'm trying to understand why in a ring we choose the names unit to an invertible element and irreducible element in this definition Maybe historical reasons? For example, I suppose the second ...