Articles of abstract algebra

Prove that a UFD is a PID if and only if every nonzero prime ideal is maximal

Prove that a UFD is a PID if and only if every nonzero prime ideal is maximal. The forward direction is standard, and the reverse direction is giving me trouble. In particular, I can prove that if every nonzero prime ideal is maximal then every maximal ideal is principal. From here, I know every ideal […]

Tensor product algebra $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$

I want to understand the tensor product $\mathbb C$-algebra $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$. Of course it must be isomorphic to $\mathbb{C}\times\mathbb{C}.$ How can one construct an explicit isomorphism?

Example of infinite groups such that all its elements are of finite order

I am in need of: Example of infinite groups such that all its elements are of finite order.

Explanation for why $1\neq 0$ is explicitly mentioned in Chapter 1 of Spivak's Calculus for properties of numbers.

During the first few pages of Spivak’s Calculus (Third edition) in chapter 1 it mentions six properties about numbers. (P1) If $a,b,c$ are any numbers, then $a+(b+c)=(a+b)+c$ (P2) If $a$ is any number then $a+0=0+a=a$ (P3) For every number $a$, there is a number $-a$ such that $a+(-a)=(-a)+a=0$ (P4) If $a$ and $b$ are any […]

Relating $\operatorname{lcm}$ and $\gcd$

I would appreciate help to show this equality is valid: $\operatorname{lcm} (u, v) = \gcd (u^{- 1}, v^{- 1})^{- 1}$, where $u, v$ are elements of a field of fractions. In the text it is stated that lcm is: there is an element $m$ in K for which $u| x$ and $v| x$ is equivalent […]

The p-adic numbers as an ordered group

So I understand that there is no order on the field of p-adic numbers $\mathbb{Q_p}$ that makes it into an ordered field (i.e.) compatible with both addition and multiplication. Now, from the responses to a couple of my previous questions, $\mathbb{Q_p}$ is a divisible abelian group under addition (being a field of characteristic $0$). $\mathbb{Q_p}$ […]

When is a tensor product of two commutative rings noetherian?

In particular, I’m told if $k$ is commutative (ring), $R$ and $S$ are commutative $k$-algebras such that $R$ is noetherian, and $S$ is a finitely generated $k$-algebra, then the tensor product $R\otimes_k S$ of $R$ and $S$ over $k$ is a noetherian ring.

If an element has a unique right inverse, is it invertible?

Suppose $u$ is an element of a ring with a right inverse. I’m trying to understand why the following are equivalent. $u$ has at least two right inverses $u$ is a left zero divisor $u$ is not a unit If $v$ and $w$ are distinct right inverse of $u$, then $u(v-w)=0$, but $v-w\neq 0$, so […]

Determining the structure of the quotient ring $\mathbb{Z}/(x^2+3,p)$

I’m interested in the following problem from Artin’s Algebra text: Determine the structure of the ring $\mathbb Z[x]/(x^2 + 3,p)$, where (a) p = 3, (b) p = 5. I know that by the isomorphism theorems for rings we can take the quotients successively, and so $$\mathbb{Z}[x]/(p) \cong (\mathbb{Z}/p \mathbb{Z})[x] $$ as the map $\mathbb{Z}[x] […]

An ideal which is not finitely generated

Let $K$ be a field and $A=K[x_1,x_2,x_3,…]$. Prove that the ideal $I:=\langle x_i: i \in \mathbb N\rangle$ is not finitely generated as $A$-module. I have no idea what can I do here, I mean, suppose $I$ is finitely generated, then there is a subset $S \subset I$ with $S=\{x_{i_1},…,x_{i_n}\}$ such that $\langle\{x_{i_1},…,x_{i_n}\}\rangle=I$. How can I […]