Tagged Questions

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generators vs basis of an algebra

Are all bases of an algebra generating sets, but all generating sets are not bases? A basis can only use addition and scalar multiplication to generate an algebra, which means it is a generating ...
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Usage of the term “Free”

What is the difference between free groups and free modules/vector/algebras spaces, or in other words what does free mean in algebra? I have seen two uses of the term free: something of ...
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Why are normal subgroups called “Normal”?

Why are normal subgroups called "Normal"? Who is credited with naming them, and why are they named such?
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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 ...
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Why do we use the term “equivalent” with Operators but “equal” with Functions?

Why do we speak in terms of "equality" when we deal with functions but "equivalence" when dealing with operators? To elaborate: Two functions, f and g are equal to each other (denoted: f=g) if: ...
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What's the name of the set of products of equal to a given value?

Suppose we have the * operator on a set $A$ such that * is associative but not commutative. Given $a$, $b$, $c \in A$, \begin{align*} abc &= (abc) \\ &= (a)(bc) \\ &= (ab)(c) \\ &= ...
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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 ...
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Some basic terms from finite group theory, normalising and centralising

In a proof a read the following "Since $H$ and $N$ normalize each other, if follows that $H \subseteq C_G(N)$". I thought normalising just means that one subgroup is normal with respect to the other, ...
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Has this property for algebraic structures got a name?

Given a binary operation $*:M\times M\rightarrow M$. I define the extension of $*$ to the subset of $M$ in the usual way (with $A,B \subseteq M$) $a*A:=\{a\}*A$and $A*a:=A*\{a\}$, ...
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Is there a term for an algebraic structure with two binary operators that are closed under a set?

For example, let's say we're using the operators +, and *, and the set {0,1,2} The Cayley tables look like this: ...
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What are the terms for the elements in the Euclidean algorithm $a = qb + r$?

In a ring $R$ with a Euclidean norm $N$, given $a,b \in R$ with $b\neq 0$, there are elements $q, r$ such that $a = qb + r$. Is there any special terminology for the elements $a,b,q,r$ in this ...
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Why are groups “abelian” but rings “commutative”?

I have never seen, in any text, a ring whose multiplication is commutative being called an "abelian ring", even though this would make perfect sense, because this term would necessarily refer to ...
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“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 ...
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Two natural extensions of every algebra. Extension to subsets or functions.

I don't exactly know the technical meaning of extension, but I was thinking that given a set $A$ and an operation $*$ on it we can extend the set $A$ in a very natural way and thus extend any ...
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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 ...
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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 ...
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Terminology with zero divisors

Let $R$ be a commutative ring with identity $1$. If for some $a \in R$ there exists $b \in R$ such that $ab = 1$, then we say that $a$ is a unit and that $b$ is a multiplicative inverse or reciprocal ...
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What is the “opposite” of a forgetful functor?

Consider a category $C$ and a monoid $M$. Consider a functor $F:C\to M$. It maps the objects of $C$ into the only object of $M$. But I don't want it to map every morphism of $C$ into the identity on ...
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Is there a name for the generalization of the concept “Abelian group” where the axiom $-x+x = 0$ is weakened to the following?

Is there a name for the generalization of the concept "Abelian group" where the axiom $−x+x=0$ is replaced by the following list? $−0=0$ $−(x+y)=−x+−y$ $−(−x)=x$ $x+(-x)+x = x$ In multiplicative ...
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What's the name of the mathematical structure with is an abstraction of things like linear Independence?

This is a terrible question, I know. I can't remember the details for some reason, but I think (hope?) that anyone who's familiar with this object will immediately know what I'm talking about.
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Is there a name for the algebra of substructures?

Let $X$ denote an entropic algebra (see here), which just means that all the operations of $X$ are homomorphisms $X^n \rightarrow X.$ Abelian groups are the classic example. Then for any operation of ...
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Have action/predicate systems (or similar) been considered in the literature?

Question. Has the following concept, or anything similar, been considered in the literature? Definition. An action/predicate system consists of sets $A$ (the actions) and $X$ (the predicates) such ...
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Weakened associativity axiom: $(x * y) * z \leq x*(y*z).$

Call a partially ordered magma $(X,*)$ sub-associative iff it satisfies the following axiom. $$(x*y)*z \leq x*(y*z).$$ Basically, this is saying that we may shuffle brackets right to get a larger ...
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Meaning of 'Isomorphism (with respect to inclusion)'

This is the first time that I see this phrase. I'm reading Commutative Algebra by N.Bourbaki. I'll extract 2 propositions that use this phrase. The first one is on page 68 of the book. ...
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Do subhomomorphisms / subfunctors have a standard name, and where can I learn more?

Sorry for all the mistakes in the original! I think they're mostly fixed now. Thank you for your patience. Part 1. If $A$ and $B$ are models in $\mathrm{Pos}$ of an algebraic signature $\sigma$, ...
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The Empty Relation?

In elementary set theory, a relation on sets $A,B$ is usually defined as a subset of $A\times B$. We know that there are $2^{|A\times B|}$ subsets of $|A\times B|$. One of these subsets is the empty ...
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What is the correct usage of the terms ‘under’ and ‘over’ in abstract algebra?

For example, the Wikipedia article on rings states that a ring is: an abelian group under addition a monoid under multiplication multiplication distributes over addition
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Name for this axiom $(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$

I am trying to give a name to this axiom in a definition: $(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$ (for all $X, Y, R, S$) where $\sqcup$ is the join of a ...
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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.
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Is there a name for magmas in which $y*(a*b) = (y*a)*(y*b)$?

Let $X$ and $Y$ denote magmas, suppose $f$ is a homomorphism $X \rightarrow Y$, and let $y \in Y$ satisfy the following condition. $$\forall a,b \in Y : y*(a*b) = (y*a)*(y*b)$$ Then $y * f$ is a ...
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Taking the (pseudo)inverse of a monoid operation.

Let $M$ be a monoid with binary operation $f : M \times M \to M$. I'm interested in functions $g : M \to M\times M$ that obey the property: $$f(g(m)) = m$$ I want to understand what all of the ...
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Terminology: How should we call $\mathbb{Z}[\sqrt{5}]$?

I'm wondering, what shall we call the ring $\mathbb{Z}[\sqrt{5}]$? I know that $\mathbb{Z} \left[ \frac{1+\sqrt{5}}{2} \right]$ is called a quadratic integer ring. But do we have something similar for ...
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Is there a name for magmas with $[x+y]+[x'+y'] \equiv [x+x']+[y+y']$?

Is there a name for magmas (written additively) satsisfying the following identity? The square brackets have no particular signifance, but will hopefully promote readability in what follows. ...
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Why are group theory and ring theory a part of abstract algebra?

I have followed the courses Algebra 1, which was about group theory and Algebra 2, which was about ring theory. I don't think I really understand why those subjects are part of abstract algebra. What ...
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Terminology regarding elements of monoids

In what follows, the symbols $a,b$ and $n$ implicitly range over $\mathbb{N} = \{0,1,2,\cdots\}.$ Are there names for the following properties that an element $x$ in a monoid may or may not possess? ...
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A module as an external direct product of the kernel and image of a function

If $f:A\rightarrow A$ is an R-module homomorphism such that $ff=f$, show that $$A=Ker\,\,f\oplus Im\,\, f$$ Here is a part of what I made as a proof. Let $a\in A$. $f(a)\in Im\,\,f$, ...
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First Homomorphism Theorem and terminology

I am not finding clear terminology in my abstract algebra book to be clear at least and my questions are simple. Consider the construction of a quotient group G': G/K = G' ...
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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 ...
I'm independently studying abstract algebra for fun (not my forte...) and I'm reading Herstein. He has a question in the chapter on rings: Let $p$ be an odd prime and ...
When we study actions in group theory, we consider sets of the form $$\text{Orb}_G(x)=\{gx\mid g\in G\}$$ that are called orbits. Although, the only reason I find convincing for that name is that in ...