# Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc as necessary. To clarify which topic of abstract algebra is most related to your question and help other users when searching.

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### Natural morphism/s: The morphism doesn't preserve identity elements.

$r \mapsto (r,0)$ is the natural map 1 $r \mapsto (r,0)$ is the natural map 2 Considering the two rings $R$ and $R \times R$ the ring morphism $r \mapsto (r,0)$ is not a ring homomorphism, but it's ...
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• 1,277
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### Every principal ideal domain is a dedekind domain.

I thought about principal ideal domains (PID) and dedekind domains, and something confuses me a bit. I don't know if I have a thinking error. I know that every principal ideal domain is a dedekind ...
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### Maximal and prime ideals in $\mathbb{Z}[x]$

I am checking if the ideals $$I=(5,x^3+2x+3), \qquad J=(4,x^2+x+1,x^2+x-1) \trianglelefteq \mathbb{Z}[x]$$ are either maximal or prime. For the ideal $I$, by factorizing $x^3+2x+3=(x+1)(x^2-x+3)$ I ...
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### Is one-to-one-ness and operation preservation sufficient to prove an isomorphism between two finite groups with identical cardinality?

See question. In my Abstract Algebra textbook, it is mentioned that to prove an isomorphism $\phi$ exists is to show it is well defined, one-to-one, onto, and operation preserving. It seems like ...
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### about finite group with order $100$.

let $G$ be a finite group with order $100$ and $H$ is a subgroup of $G$ with order $25$ also let $a \in G$ has order $50$. Now which of following options is true? 1)$\quad$ $|\langle a \rangle H |=50$ ...
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### What does a 3-tuple mean as notation for a group?

$(F, +, 0)$ I am looking at my linear algebra lecture notes for next year and immediately came upon this notation I've never seen before, in the context of defining a field. Clearly $F$ is the set, $+$...
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### Doubt about degree for polynomimal over field?

What is the mistake of the following case:Suppose $\mathbb{F}=\mathbb{Z}_p$ and $p$ is prime.Then $x^p\equiv x$ for all $x$,then we derive $p=\text{deg}(x^p)=\text{deg}(x)=1$?
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### Natural partial order on idempotents and $\mathcal D$ relation

The natural partial order on the idempotents of a semigroup is defined by $$e \leq f \; \; \text {iff} \;\; ef = fe = e$$ My question regards idempotents and Green's relations: Can two different ...
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### Showing that the additive group on the dyadic rationals is not a free Abelian group [duplicate]

I'm wondering whether $(\mathbb{D}, +)$ the additive group of the dyadic rationals is isomorphic to a free Abelian group or not. On an intuitive level, it seems like it's obviously nonfree, but I'm ...
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### Prove $\mbox{adj} (A) = \left[ \frac{\partial }{\partial a_{ij} } \det(A) \right]^T$

I have a bit of trouble proving an adjugate matrix equality for $A_{n\times n} = [a_{ij}]$, $$\mbox{adj} (A) = \left[ \frac{\partial }{\partial a_{ij} } \det(A) \right]^T, \quad i, j = 1,\dots,n$$ ...
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### $k$ field with characteristic $p$, polynomial $x^{p^n} - 1$ obviously has a single root, what about $x^{pn} - 1$?

Lang's Algebra (VI. 3. Roots of unity); Lang actually never mentions this as far as I know: If $k$ is a field with characteristic $p$, then the polynomial $x^{p^n} -1$ has only one root and that's 1, ...
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