Articles of ring theory

Ring of rational numbers with odd denominator

Consider the ring $R$ of rational numbers which, when written in simplest form, have an odd denominator. This is a subring of $\mathbb{Q}$ with the usual multiplication and addition. I wonder if you could check my understanding of the units, irreducible elements, and prime elements in this ring. Assume all fractions are in simplest form. […]

Identify the ring $\mathbb{Z}/(x^2-3, 2x+4)$

This question already has an answer here: Cardinality of the quotient ring $\mathbb{Z}[x]/(x^2-3,2x+4)$ [duplicate] 3 answers

Two problems about rings.

Somebody can to help me in such exercices: (1) A ring R such that $a^2 = a$ for all $a\in R$ is called a Boolean ring. Prove that every Boolean ring R is commutative and $a + a = 0$ for all $a \in R$. (2) Let R be a ring with more than one […]

Prime and Primary Ideals in Completion of a ring

Let $(R,\mathfrak m)$ be a local noetherian ring and $\widehat{R}$ its $\mathfrak m$-adic completion. If $\mathfrak q\in \operatorname{Spec}(\widehat{R})$ then can we find $\mathfrak p\in \operatorname{Spec}(R)$ such that $\mathfrak q=\mathfrak p\widehat{R}$ ? Is a primary ideal of $\widehat{R}$ an extension of a primary ideal of $R$ ?

Is the ring of germs of $C^\infty$ functions at $0$ Noetherian?

I’m considering the property of the ring $R:=C^\infty(\mathbb R)/I$, where $I$ is the ideal of all smooth functions that vanish at a neighborhood of $0$. I find that $R$ is a local ring of which the maximal ideal is exactly $(x)$. I also want to determine if $R$ is a Noetherian ring, but I have […]

Sufficient and necessary conditions for $f,g$ such that $\Bbb F_m/(f) \cong \Bbb F_m/(g)$

Consider the polynomial ring $\Bbb F_m[x]$, and two polynomials $f,g $ in $\Bbb F_m[x]$. Is there any necessary and sufficient conditions for $f,g$ such that $\Bbb F_m[x]/(f) \cong \Bbb F_m[x]/(g)$? Or if there are two particular $f,g$ in $\Bbb F_m[x]$, how to check that the quotient rings generated by $f,g$ are isomorphic? I have this […]

Maximal ideal in $K$

This question already has an answer here: Maximal ideals in $K[X_1,\dots,X_n]$ 5 answers

Proving ring $R$ with unity is commutative if $(xy)^2 = x^2y^2$

I tried several methods to solve this but couldn’t get through. Now the solution in almost all the textbooks goes like this. First take $x$ and $y+1$ so that $ (x(y+1))^2 = x^2(y+1)^2 => xyx = x^2y \\$ Then substitute $x+1$ in place of $x$ to get $yx = xy$ I have always believed that […]

associative and commutative F-algebra with basis

let $G$ be a finite-dimensional $F$-algebra and let $X=\{x_1,…,x_m\}$ be the basis. The structure constants of $G$ with respect to the basis are the scalars $f_{rsk}\in F$ for $1\leq r,s,k\leq m$ given $x_rx_s= f_{rs1}x_1+…+f_{rsm}x_m$. What are the conditions on the structure scalars $f _{rsk}$ that are equivalent to associativity and commutativity for algebra $G$?

global dimension of rings and projective (flat) dimension of modules

Let $R$ be ring such that every left $R$-module has finite projective dimension ( resp. finite injective dimension). Is the left global dimension of $R$ finite? Similarly, Let $R$ be ring such that every left $R$-module has finite flat dimension. Is the weak global dimension of $R$ finite?