Articles of abstract algebra

Prove that if R is a PID, then R is a field

I just need someone to check my proof and provide me feedback: Since $R[x]$ is a PID, then the ideal $I = (x-1)$ generated by the polynomial $x-1$ is maximal because it is of degree 1 added to a constant. So $R[x]/I\simeq R$ is a field, so $R$ is a field. Is this proof enough/correct? […]

Name for a semiring minus multiplicative identity requirement

Is there a name for a theory that has all axioms of a semiring but an axiom that mandates multiplicative identity?

What is the “circle-plus” symbol I see in Abstract Algebra?

Sorry if this comes off as a random or soft question. I keep seeing this symbol in my abstract algebra course where it is a plus sign inside of a circle. I am not sure what it means. Can someone please help me? Sorry if this bothers anyone. Thanks.

Galois group over the field of rational functions

I am looking to find the Galois group of $x^3-x+t$ over $\mathbb{C}(t)$, the field of rational functions with complex coefficients. I have shown that the automorphisms of the rational function field $F(t)$ for fixed $F$ are precisely the fractional linear transformations that is $t \rightarrow \frac{at +b}{ct+d}$ for $a,b,c,d \in \mathbb{C}$. Is this useful? Also […]

Proving Finiteness of Group from Presentation

Given the group $G = \langle a, b, c : a^2 = b^3 = c^5 = abc\rangle$, I want to show that $H = G / \langle abc\rangle$ is a finite group. I tried to find a canonical form for elements of $H$. That didn’t work. Then I tried to find a finite normal subgroup […]

Question about a projective module in a direct sum

I have a question about my method of proof for proving a simple fact about projective modules. I have a feeling my idea is wrong and I was hoping some one could point out where the mistake is. Let $R$ be a commutative ring with identity. Let $P$ be a projective $R$-module. Prove that there […]

Von Dyck's theorem (group theory)

Did anyone find a proof of this theorem? I can’t find it on the Internet. The theorem is : Let $X$ be a set and let $R$ be a set of reduced words on $X$. Assume that a group $G$ has the presentation $\langle X \mid R \rangle$. If $H$ is any group generated by […]

Confusing partitions of $S_5$ in two different sources

I am trying to understand the partitions of $S_5$ created by it’s conjugacy classes but two sources have two different partitions. Source 1: Source 2: So, for example, in the first table, the partition for cycle structure ()()()()() i.e. $5$ $1$-cycle is $5+0+0+0+0$ but in the second table it is $1+1+1+1+1$. Could anyone please help […]

Where is the “relation” here?

Looking at the definition of a monoid it says that: A monoid is a set that is closed under an associative binary operation and has an identity element $I \in S$ such that for all $a \in S$, $I a = a I =a$ But what does $I a$ mean here? I mean it’s just […]

Showing isomorphism between two rings of fractions

Take $R = \mathbb{Z}$ and fix a set of nonzero integers $X = \{n_{1},n_{2},\dots, n_{k} \}$. Now, let $$D = \{n_{1}^{t_{1}}n_{2}^{t_{2}}\cdots n_{k}^{t_{k}} \mid t_{i}\geq 0 \, \text{for all}\, i,\ t_{i}\neq 0\, \text{for at least one}\, i\}$$ i.e., $D$ is a semisubgroup generated by $X$. I need to show that $D^{-1}\mathbb{Z}$ is isomorphic to another ring […]