Articles of abstract algebra

Show $M=0\iff M_{\mathfrak p}=0\iff M_{\mathfrak m}=0$.

Let $R$ a commutative ring and $M$ a $R-$module. Show that the following statement are equivalent: 1) $M=0$, 2) $M_{\mathfrak p}=0$ for all $\mathfrak p\in Spec(R)$, 3) $M_{\mathfrak m}=0$ for all $\mathfrak m\in Specm(R)$. Proof : $\bullet1)\Rightarrow 2)$: Let $M=0$. Let $\frac{a}{b}\in M_{\mathfrak p}$ (i.e. $b\notin \mathfrak p$). Then, $$\frac{a}{b}=\frac{a}{1}\frac{1}{b}=\frac{0}{1}$$ since $\frac{a}{1}=\frac{0}{1}$ and $\frac{0}{1}\frac{1}{b}=0$. $\bullet […]

Automorphism on integers

Is multiplying by a constant m (integer) on group of set of all integers on addition an automorphism? If so why does the 2nd example in says that the unique non trivial automorphism is negation?

What is the relationship between the second isomorphism theorem and the third one in group theory?

The second isomorphism theorem [wiki] in group theory is as follows: Let $G$ be a group. $H \triangleleft G, K \le G$. Then: $HK \le G$, $(H \cap K) \triangleleft K$, and $K/(H \cap K) \cong HK/H \le G/H$. The third isomorphism theorem [wiki] is as follows: Let $G$ be a group. $K \triangleleft G$. […]

Is $R$ a necessarily UFD for $X$ an infinite set of symbols?

Let $R$ denote a UFD, and let $X = \{x_0,\cdots,x_{n-1}\}$ denote a finite set of symbols. Then $R[X]$ is a UFD. This follows, since if $R$ is a UFD, then so too is $R[x],$ for any symbol $x \notin R$. Q. If $R$ is a UFD, and $X$ is an arbitrary set, possibly infinite, is […]

Isomorphism from $B/IB$ onto $(B/I)$

For some reason I can’t crack the following problem: Let $B$ be a ring, $I$ an ideal, and $A := B[y]$ the polynomial ring. Construct an isomorphism from $A/IA$ onto $(B/I)[y]$. How to interpret the (polynomial) quotient ring $A/IA$? And how do I construct the isomorphism knowing this? Hope you can help!

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) […]

Norm of Gaussian integers: solutions to $N(a) = k$ for $k \in \mathbb{N}$ and $a$ a Guassian integer

I have two questions. Which integers are equal to the norm of some Gaussian integer? In general, how many solutions does$$\text{N}(a) = k$$have for a given $k \in \mathbb{Z}$? I am investigating the equation$$\text{N}(a) = 1$$where now we take $a \in \mathbb{Q}(i)$. What is a method for producing solutions to this equation using the arithmetic […]

Criterion for isomorphism of two groups given by generators and relations

When are two presentations of groups are isomorphic? In this post it is said: […] find a set of generators of the first group that satisfies the relations of the second group […] But I doubt that, as this just shows that the first group is an epimorhpic image of the second group. For example […]

Properties of automorphism group of $G={Z_5}\times Z_{25}$

Let $G={\Bbb Z}_5\times {\Bbb Z}_{25}$. What is the order of $Aut(G)$ and is $Aut(G)$ abelian? I don’t know if this has anything to do with the result $$ Aut({\Bbb Z}_n)\cong U(n). $$ Does one have to count $Aut(G)$ by “brute force”? I’ve seen two similar questions: number of automorphism on $\mathbb{Z}_9\times \mathbb{Z}_{16}$ Order of the […]

Every infinite abelian group has at least one element of infinite order?

Is the statement true? Every infinite abelian group has at least one element of infinite order. I am searching for an infinite abelian group with all elements having finite order. Please help me to find such groups.