Tagged Questions

This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

1answer
56 views

Determining if (3) is a maximal ideal in $\mathbb{Z}[\sqrt{7}]$.

As far as I can tell, the tools I have for determining if an ideal I of a ring R is maximal is either: Determine another ideal it is contained within, or look at the quotient ring $R/I$ and determine ...
2answers
142 views

Left ideals of matrix rings are direct sum of column spaces?

Let $\mathbb K$ be a field and $M_n(\mathbb K)$ be the ring of the $n\times n$ matrices with entries in $\mathbb K$. Let $C_j\subset M_n(\mathbb K)$ be the subspace of all matrices which have all ...
1answer
23 views

Show that $a$ is irreducible iff $au$ is irreducible where $u$ is invertible

$R$ is integral domain. Show that $a$ is irreducible iff $au$ is irreducible where $u\in R^*$. My Try: Lets assume $a$ is irreducible and $au = bc \implies a=bcu^{-1}$. We know that $u\in R^*$ so we ...
2answers
33 views

$R$ integral domain : $u\in R^*, a \text{ is prime} \iff au \text{ is prime}$

$R$ integral domain : $u\in R^*,\; a \text{ is prime} \iff au \text{ is prime}$ I started by looking at $auu^{-1}$. What should I do next? I'd be glad for help. Note: $u \in R^*$ meaning is $u$ is ...
1answer
63 views

1answer
56 views

Is a ring $R$ factorial $\iff$ $R[X]$ factorial?

Let $R$ be a factorial ring. Then, the polynomial ring $R[X]$ is factorial. I was wondering if the other direction also works (i.e. $R[X]$ factorial $\implies$ $R$ factorial)? If not, please give ...
2answers
39 views

Property of unfaithful module over PID

Proposition Let $R$ be a PID and $M$ a torsion module over $R$. Suppose $M$ is not faithful, with $\operatorname{Ann(M)}=Ra$ and $a=u{p_1}^{\alpha_1}...{p_n}^{\alpha_n}$ an irreducible factorization ...
2answers
136 views

Determine whether the polynomial $x^2-12$ in $\mathbb Z[x]$ satisfies an Eisenstein criterion for irreducibility over $\mathbb Q$

Determine whether the polynomial $x^2-12$ in $\mathbb Z[x]$ satisfies an Eisenstein criterion for irreducibility over $\mathbb Q$. So If I understand correctly, I start with $x^2=12$ and then you ...
1answer
107 views

2answers
178 views

Confusion with definition of irreducible.

An integer $p$ is said to be irreducible if whenever $p=ab$ then $a$ or $b$ is $1$ or $-1$. Then we define an irreducible element $p$ in a commutative ring $R$ with unity as: $1)$ $p \neq 0$ ...
1answer
65 views

If A , B are finitely generated R-algebras then $A\otimes_RB$ is a finitely generated $R$-algebra.

$A$, $B$ are finitely generated $R$-algebras. $R$ is a commutative ring with $1$. Then how can I show that $A\otimes_RB$ is finitely generated $R$-algebra? What I have tried: First I have to show ...
1answer
96 views

1answer
80 views

Is this “sliding window” unique?

Starting with $x$, which is a positive integer or zero, and $y$ a second positive integer or zero, with $y \ne x$, we can create lists. Set $p$ a prime greater than 2, $\alpha = \lfloor p/2 \rfloor$, ...
3answers
56 views

1answer
40 views

A quick question on the subring generated by a finite set

Let $R$ be a commutative unital ring and let $r_1,\ldots,r_n \in R$. Let $S$ be the unital subring of $R$ generated by $r_1,\ldots,r_n$. Let $\varphi:\mathbb{Z}[X_1,\ldots,X_n]\to R$ be the unique ...
0answers
47 views