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 http://en.wikipedia.org/wiki/Automorphism 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.