Articles of ring theory

Finite ring of sets

I have some questions about finite rings of sets and I’ll be very grateful for any help. Let E be some fixed non-empty set. Suppose we are given some finite ring of subsets of set E, i.e. some non-empty system $S \subset E$ such that $\forall A,B \in 2^E ~~ A \vartriangle B \in S$ […]

normalization of multiplicative subset of domain

I am stuck with this: Let $R$ be a domain with normalization $R’ \subset K$. Show that for every multiplicative subset $S \subset R$, the normalization of $S^{-1}R$ equals $S^{-1}R’$. How do you show this? I am not yet that into the definitions and theorems I could use to prove such a statement. Hope someone […]

Subrings of polynomial rings over the complex plane

I have the following questions: (i) must every subring of the polynomial ring in two variables over the complex plane, containing the complex plane itself, be Noetherian? (ii) Are there Noetherian rings containing the complex field which are not f.g. as a ring over the complex field (maybe the field of rational functions over the […]

Non-Examples for Krull-Schmidt-Azumaya

I am looking for some rings $A$ and finitely generated $A$-modules $M$, where the conclusion of the Krull-Schmidt-Azumaya Theorem does not hold for $M$, i.e. where $M$ can be written as direct sums of indecomposable modules in multiple ways (the summands not being unique up to order and isomorphism). To be specific, I am looking […]

Ideal of $\text{End}_k V$ has certain form.

Let $V$ be a finite-dimensional vector space over field $k$ and $R = \text{End}_k V$. How do I see that any left ideal of $R$ takes on the form $Rr$ for some suitable element $r \in R$?

Factor $55 – 88 \sqrt{-2}$ as a product of primes in $\mathbb{Z}$

To solve this problem, I let $K = \mathbb{Q}(\sqrt{-2})$, and I thought to take the norm $$N(55 – 88 \sqrt{-2}) = 55^2 + 2 \cdot 88^2 = 18513 = 3^2\cdot11^2 \cdot 17$$ If $a \in \mathbb{Z}[\sqrt{-2}]$ is irreducible, then $N(a) = p^f$, where $a$ lies over the prime $p$ and $f$ is the inertia degree […]

Semisimple ring problem

Prove that: $R$ is a semisimple ring $\Longleftrightarrow$ Every right $R$-module is injective (projective) My try: $R$ is semisimple ring $\Longleftrightarrow$ Every right $R$-module is semisimple $\Longleftrightarrow$ Every submodule is direct summand Please explain that why since every submodule is direct summand then every $R$-module is injective (projective)?

Class Group of Ring of Integers of $\mathbb{Q}$

Let $R$ denote the ring of integers of the imaginary quadratic number field $\mathbb{Q}[\sqrt{-57}]$. I must find the ideal class group $\mathcal{C}$. Using the Minkowski Bound, I know that I need only look at the primes $2,3,5,7$. Moreover, because $x^2+57 \equiv x^2 \mod 3$, $x^2+57 \equiv x^2+2 \mod 5$, $x^2+57 \equiv x^2+1 \mod 7$, it […]

Identify the structure of $\mathbb{Z}/(x^2+3,3)$

I assume that $\mathbb{Z}[x]/(x^2+3)\simeq\mathbb{Z}[i\sqrt{3}]$. Thus we should have $\mathbb{Z}[i\sqrt{3}]/(3)$? That would be smaller than $\mathbb{Z}_3[i\sqrt{3}]$ because $3\in(i\sqrt{3})$? If so, what would be such a ring? Would there be more straight-forward approach?

Find all elements of quotient ring

I am studying the definitions of rings, ideals, and quotient ring, but I have a bit problem to apply the theory into the practice. I would like to find all elements of quotient ring $\mathbb{Z}[i]/I $ , where $\mathbb{Z}[i] =\{ {a+bi|a, b∈ \mathbb{Z}}\}$ – Gaussian integers and $I$ is ideal $I = (2 + 2i) […]